A note on the link Floer homology of doubly-periodic knots
Kristen Hendricks
A knot $\widetilde{K} \subset S^3$ is $q$-periodic if there is a $\mathbb Z_q$-action preserving $\widetilde{K}$ whose fixed set is an unknot $U$. The quotient of $\widetilde{K}$ under the action is a second knot $K$. We construct equivariant Heegaard diagrams for $q$-periodic knots, and show that Murasugi's classical condition on the Alexander polynomials of periodic knots is a quick consequence of these diagrams. For $\widetilde{K}$ a two-periodic knot, we show there is a spectral sequence whose $E^1$ page is $\widehat{\mathit{HFL}}(S^3,\widetilde{K}\cup U)\otimes V^{\otimes (2n-1)})\otimes \mathbb Z_2((\theta))$ and whose $E^{\infty}$ pages is isomorphic to $(\widehat{\mathit{HFL}}(S^3,K\cup U)\otimes V^{\otimes (n-1)})\otimes \mathbb Z_2((\theta))$, as $\mathbb Z_2((\theta))$-modules, and a related spectral sequence whose $E^1$ page is $(\widehat{\mathit{HFK}}(S^3,\widetilde{K})\otimes V^{\otimes (2n-1)}\otimes W)\otimes \mathbb Z_2((\theta))$ and whose $E^{\infty}$ page is isomorphic to $(\widehat{\mathit{HFK}}(S^3,K)\otimes V^{\otimes (n-1)} \otimes W)\otimes \mathbb Z_2((\theta))$. As a consequence, we use these spectral sequences to recover a classical lower bound of Edmonds on the genus of $\widetilde{K}$, along with a weak version of a classical fibredness result of Edmonds and Livingston.
PeriodicKnotsPaperHendricks.tex
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\author{Kristen Hendricks}
\thanks{The author was partially supported by an NSF grant number DMS-0739392.}
\title{A note on the link Floer homology of doubly-periodic knots}

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\begin{abstract}

A knot $\widetilde{K} \subset S^3$ is $q$-periodic if there is a $\mathbb Z_q$-action preserving $\widetilde{K}$ whose fixed set is an unknot $U$.  The quotient of $\widetilde{K}$ under the action is a second knot $K$.  We construct equivariant Heegaard diagrams for $q$-periodic knots, and show that Murasugi's classical condition on the Alexander polynomials of periodic knots is a quick consequence of these diagrams. For $\widetilde{K}$ a two-periodic knot, we show there is a spectral sequence whose $E^1$ page is $\widehat{\mathit{HFL}}(S^3,\widetilde{K}\cup U)\otimes V^{\otimes (2n-1)})\otimes \mathbb Z_2((\theta))$ and whose $E^{\infty}$ pages is isomorphic to $(\widehat{\mathit{HFL}}(S^3,K\cup U)\otimes V^{\otimes (n-1)})\otimes \mathbb Z_2((\theta))$, as $\mathbb Z_2((\theta))$-modules, and a related spectral sequence whose $E^1$ page is $(\widehat{\mathit{HFK}}(S^3,\widetilde{K})\otimes V^{\otimes (2n-1)}\otimes W)\otimes \mathbb Z_2((\theta))$ and whose $E^{\infty}$ page is isomorphic to $(\widehat{\mathit{HFK}}(S^3,K)\otimes V^{\otimes (n-1)} \otimes W)\otimes \mathbb Z_2((\theta))$. As a consequence, we use these spectral sequences to recover a classical lower bound of Edmonds on the genus of $\widetilde{K}$, along with a weak version of a classical fibredness result of Edmonds and Livingston.


\end{abstract}

\maketitle

\section{Introduction}

We say $\widetilde{K} \subset S^3$ is a \textit{periodic knot} if there is a $\mathbb Z_q$-action on $(S^3, \widetilde{K})$ which preserves $\widetilde{K}$ and whose fixed set is an unknot $U$ disjoint from $\widetilde{K}$.  Let $\tau$ be a generator for the action.  An important special case is $\tau^2=1$, in which case $\widetilde{K}$ is said to be \textit{doubly-periodic}.  

\medskip

The quotient of $(S^3, \widetilde{K})$ under the group action is a second knot $(S^3, K)$ such that the map $(S^3, \widetilde{K}) \rightarrow (S^3,K)$ is an $q$-fold branched cover over $U$.  The knot $K$ is said to be the $q$-fold \textit{quotient knot} of $\widetilde{K}$.

\medskip

\begin{figure}
\caption{A doubly-periodic diagram for the trefoil, and its quotient knot (an unknot) under the $\mathbb Z_2$ action.}
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Link Floer homology is an invariant of a link in $S^3$ introduced by Ozsv{\'a}th and Szab{\'o} \cite{MR2443092} as a generalization of the knot Floer homology developed in 2003 by Ozsv{\'a}th and Szab{\'o} \cite{MR2065507} and independently by Rasmussen \cite{MR2704683}.  The theory associates to $(S^3,L)$ a bigraded $\mathbb F_2$-vector space $\widehat{\mathit{HFL}}(S^3,L)$ which arises as the homology of the Floer chain complex of two Lagrangian tori in the symmetric product of a punctured Heegaard surface for $(S^3,L)$.  The graded Euler characteristic of link Floer cohomology is the multivariable Alexander polynomial of the knot multiplied by certain standard factors \cite{MR2443092}; indeed, the theory categorifies the Thurston norm of the knot complement \cite{MR2393424}.

\medskip

In this paper, we will analyze the link Floer homology of a periodic (usually doubly-periodic) knot together with its axis compared to the link Floer homology of the quotient knot together with the axis.  We will first construct Heegaard diagrams for $(S^3, \widetilde{K} \cup U)$ which are preserved by the action of $\mathbb Z_q$ and  whose quotients under the action are Heegaard diagrams for $(S^3,K)$. These periodic Heegaard diagrams will allow us to give a simple Heegaard Floer reproof of one of Murasugi's conditions for the Alexander polynomial of a periodic knot in the case that $q=p^r$ for some prime $p$. Let $\lambda = \ell k(\widetilde{K}, U) = \ell k(K,U)$.


\begin{theorem} \label{Murasugi Theorem} \cite[Corollary 1]{MR0292060} $\Delta_{\widetilde{K}}(t) \equiv  t^{\pm i}(1 + t + \cdots + t^{\lambda-1})^{q-1} (\Delta_{K}(t))^{q}$ modulo $p$.
\end{theorem}

Restricting to the case that $\widetilde{K}$ is doubly periodic, we will proceed to prove the following localization theorem. Let $\mathbb Z_2((\theta)) = \mathbb Z_2[[\theta]](\theta^{-1})$.

\begin{theorem} \label{Link Floer Homology Spectral Sequence}

There is an integer $n_1$ less than half the number of crossings of a periodic diagram $D$ for $\tilde{K}$ such that there is a spectral sequence whose $E^1$ page is $\left(\widehat{\mathit{HFL}}(S^3, \widetilde{K} \cup U) \otimes V^{\otimes (2n_1-1)}\right) \otimes \mathbb Z_2((\theta))$ and whose $E^{\infty}$ page is isomorphic to $\left(\widehat{\mathit{HFL}}(S^3, K \cup U) \otimes V^{\otimes (n_1-1)}\right) \otimes  \mathbb Z_2((\theta))$ as $\mathbb Z_2((\theta))$-modules.

\end{theorem}

We shall see that this spectral sequence splits along the Alexander multigrading of $\widehat{\mathit{HFL}}(S^3, \widetilde{K} \cup U) \otimes V^{\otimes (2n_1-1)}$, which will have a simple relationship to the Alexander multigrading of $\widehat{\mathit{HFL}}(S^3, K \cup U)\otimes V^{\otimes (n_1-1)}$.  

\medskip

We may also reduce the spectral sequence of Theorem \ref{Link Floer Homology Spectral Sequence} to contain only the information of knot Floer homology of $\widetilde{K}$ and $K$. Below, $V$ and $W$ are both two-dimensional vector spaces over $\mathbb F_2$, which will later be distinguished by their gradings.


\begin{theorem} \label{Knot Floer Homology Spectral Sequence}

There is an integer $n_1$ less than half the number of crossings on a periodic diagram $D$ for $\widetilde{K}$ such that there is a spectral sequence whose $E^1$ page is $\left(\widehat{\mathit{HFK}}(S^3, \widetilde{K}) \otimes V^{\otimes (2n_1-1)} \otimes W \right)\otimes \mathbb Z_2((\theta))$ and whose $E^{\infty}$ page is isomorphic to $\left(\widehat{\mathit{HFK}}(S^3, \widetilde{K}) \otimes V^{\otimes (n_1-1)} \otimes W\right) \otimes \mathbb Z_2((\theta))$.

\end{theorem}


This spectral sequence splits along Alexander gradings of the theory $\widehat{\mathit{HFK}}(S^3, \widetilde{K}) \otimes V^{\otimes (2n_1-1)} \otimes W$, which are related to the Alexander gradings of $\widehat{\mathit{HFK}}(S^3, \widetilde{K}) \otimes V^{\otimes (n_1-1)} \otimes W$ by division by two and an overall grading shift. Analysis of the behavior of this grading yields a reproof of a classical result, proven by Alan Edmonds using minimal surface theory.

\begin{corollary} \label{Genus Inequality Corollary} \cite[Theorem 4]{MR769284} Let $\widetilde{K}$ be a doubly-periodic knot in $S^3$ and $K$ be its quotient knot.  Then
%
\[
g(\widetilde{K}) \geq 2g(K) + \frac{\lambda-1}{2}.
\]


\end{corollary}

We also observe from the spectral sequence a proof of the following corollary.

\begin{corollary}  \label{Fiberedness Corollary}

Let $\widetilde{K}$ be a doubly-periodic knot in $S^3$ and $K$ its quotient knot.  If Edmonds' 
condition is sharp and $\widetilde{K}$ is fibered, $K$ is fibered.


\end{corollary}

\medskip

This is a weaker version of a corollary proved by Livingston and Edmonds.

\medskip

\begin{corollary}\cite[Prop. 6.1]{MR728451} Let $\widetilde{K}$ be a doubly-periodic knot in $S^3$ and $K$ its quotient knot.  If  $\widetilde{K}$ is fibered, $K$ is fibered.

\end{corollary}


The spectral sequences of Theorems \ref{Link Floer Homology Spectral Sequence} and \ref{Knot Floer Homology Spectral Sequence} are analogs of the spectral sequence for the knot Floer homology of double branched covers of knots in $S^3$ constructed in \cite{Hendricks}; their construction requires only slightly more technical complexity.  As in that paper, the key technical tool in the proofs of Theorems \ref{Link Floer Homology Spectral Sequence} and \ref{Knot Floer Homology Spectral Sequence}  is a result of Seidel and Smith concerning equivariant Floer cohomology. Let $M$ be an exact symplectic manifold, convex at infinity, containing exact Lagrangians $L_0$ and $L_1$ and equipped with an involution $\tau$ preserving $(M, L_0, L_1)$. Let $(M^{\text{inv}}, L_0^{\text{inv}}, L_1^{\text{inv}})$ be the submanifolds of each space fixed by $\tau$. Then under certain stringent conditions on the normal bundle $N(M^{\text{inv}})$ of $M^{\text{inv}}$ in $M$, there is a rank inequality between the Floer cohomology $\mathit{HF}(L_0,L_1)$ of the two Lagrangians $L_0$ and $L_1$ in $M$ and the Floer cohomology $\mathit{HF}(L_0^{\text{inv}}, L_1^{\text{inv}})$ of $L_0^{\text{inv}}$ and $L_1^{\text{inv}}$ in $M^{\text{inv}}$. More precisely, they consider the normal bundle $N(M^{\text{inv}})$ to $M^{\text{inv}}$ in $M$ and its pullback $\Upsilon(M^{\text{inv}})$ to $M^{\text{inv}} \times [0,1]$. We ask that $M$ satisfy a $K$-theoretic condition called \textit{stable normal triviality} relative to two Lagrangian subbundles over $L_0^{\text{inv}} \times \{0\}$ and $L_1^{\text{inv}} \times \{1\}$. Seidel and Smith prove the following.

\begin{theorem} \label{SeidelSmith} \cite[Section 3f]{MR2739000}
If $\Upsilon(M^{\text{inv}})$ carries a stable normal trivialization, there is a spectral sequence whose $E^1$ page is $\mathit{HF}(L_0,L_1) \otimes \mathbb Z_2((\theta))$ and whose $E^{\infty}$ page is isomorphic to $\mathit{HF}(L_0^{\text{inv}}, L_1^{\text{inv}}) \otimes \mathbb Z_2((\theta))$ as $\mathbb Z_2((\theta))$ modules. 
\end{theorem}

This paper is organized as follows: In Section 2 we recall the construction and important properties of link Floer homology.  In Section 3 we construct equivariant Heegaard diagrams for periodic knots, prove Theorem \ref{Murasugi Theorem} from these diagrams, and explain how Corollaries \ref{Genus Inequality Corollary} and \ref{Fiberedness Corollary} follow from Theorems \ref{Link Floer Homology Spectral Sequence} and \ref{Knot Floer Homology Spectral Sequence}.  In Section 4 we review the basics of Lagrangian Floer cohomology and Seidel and Smith's localization theory.  In Section 5 we check the basic symplectic geometry conditions of Seidel and Smith's theory on the symmetric products used in the computation of knot and link Floer homology, and provide a description of the homotopy type and cohomology of the most general of these symmetric products (which contains the symmetric products used to compute knot and link Floer homology as submanifolds).  In Section 6 we give a proof that this general symmetric product carries a stable normal trivialization, which will imply that the symmetric products used in the computation of knot and link Floer homology do as well.  In Section 7 we compute the spectral sequences of Theorems \ref{Link Floer Homology Spectral Sequence} and \ref{Knot Floer Homology Spectral Sequence} for the unknot and the trefoil as doubly-periodic knots as examples.


\subsection{Acknowledgements}

I am grateful to Robert Lipshitz for suggesting this problem, providing guidance, and reading a draft of this paper.  Many thanks also to Matthew Hedden, Tye Lidman, Dylan Thurston, Allison Gilmore, Jen Hom and Ciprian Manolescu for helpful conversations, and to Adam Levine for pointing out the argument of Lemma \ref{Generator Matching Lemma}.  I am also indebted to Chuck Livingston and Paul Kirk for their enthusiasm and commentary, particularly concerning the arguments leading up to Lemmas \ref{Alexander Gradings Link Lemma} and \ref{Alexander Gradings Knot Lemma}.




\section{Heegaard Floer Homology Theories} \label{Heegaard Floer Background Section}

We pause to recall the construction of link Floer homology in the three sphere, first defined by Ozsv{\'a}th and Szab{\'o} in \cite{MR2443092}.  All work is done over $\mathbb F_2$.

\begin{definition}
A \textit{multipointed Heegaard diagram} $\mathcal D = (S, \boldsymbol \alpha, \boldsymbol \beta, {\bf w}, {\bf z})$ consists of the following data.

\begin{itemize}
\item An oriented surface $S$ of genus $g$. 
\item Two sets of basepoints ${\bf w} = (w_1,\cdots,w_{n})$ and ${\bf z} = (z_1,\cdots,z_{n})$ 
\item Two sets of closed embedded curves $\boldsymbol \alpha = \{\alpha_1,\cdots,\alpha_{g+n-1}\}$ and $\boldsymbol \beta = \{\beta_1,\cdots,\beta_{g+n-1}\}$ such that each of $\boldsymbol \alpha$ and $\boldsymbol \beta$ spans a $g$-dimensional subspace of $H_1(S)$, $\alpha_i \cap \alpha_j = \emptyset = \beta_i \cap \beta_j$ for $i\neq j$, each $\alpha_i$ and $\beta_j$ intersect transversely, and each component of $S - \cup \alpha_i$ and of $S - \cup \beta_i$ contain exactly one point of ${\bf w}$ and one point of ${\bf z}$. 

\end{itemize}

\end{definition}


We use $\mathcal D$ to obtain an oriented 3-manifold $Y$ by attaching two-handles to $S \times I$ along the curves $\alpha_i \times \{0\}$ and $\beta_i \times \{1\}$ and filling in $2n$ three-balls to close the resulting manifold. This yields a handlebody decomposition $Y = H_{\boldsymbol \alpha} \cup_{S} H_{\boldsymbol \beta}$ of $Y$. For this paper we will in fact restrict to the case that this produces the three-sphere $S^3$.  The Heegaard diagram $\mathcal D$ furthermore determines a knot or link in $Y$: connect the $z$ basepoints to the $w$ basepoints in the complement of the curves $\alpha_i$, push these arcs into the handlebody $H_{\boldsymbol \alpha}$, then connect the $w$ basepoints to the $z$ basepoints in the complement of the curves $\beta_i$ and push these arcs into the $H_{\boldsymbol \beta}$ handlebody. 

\medskip

We insist on numbering our basepoints such that if $L = K_1 \cup \cdots \cup K_{\ell}$ is the link thus produced, there are $n_j$ pairs of basepoints on $K_j$, and there are integers $-1 =k_0<k_1<\cdots<k_{\ell} =n$ with $k_j - k_{j-1} = n_j$ such that $w_{k_{j-1}+1},\cdots,w_{k_j}, z_{k_{j-1}+1},\cdots,z_{k_{j}}$ are the basepoints on $K_j$.  (In the examples we are actually interested in, we will have $L = K_1 \cup K_2$ with $n_1$ pairs of basepoints on $K_1$ and a single pair of basepoints on $K_2$, so the notation will not be too bad.)

\medskip

There is an important collection of two-chains on the surface $S$ on which we shall impose one more technical condition.

\begin{definition}
A \textit{periodic domain} is a 2-chain $P$ on $S \backslash \{{\bf w}\}$ whose boundary may be expressed as a linear combination of the $\alpha$ and $\beta$ curves. 
\end{definition}

Note that this definition agrees with the convention of \cite[Definition 3.4]{MR2443092}, in which the set of periodic domains is the set of 2-chains with boundary a linear combination of $\alpha$ and $\beta$ curves which contain the components of $S - {\boldsymbol \alpha} - {\boldsymbol \beta}$ containing a basepoint algebraically zero times.  The set of periodic domains on $S$ is in bijection with $\mathbb Z^{b_2(Y) + n-1} = H_2(Y \# (S^1 \times S^2)^{\#(n-1)})$. If we additionally puncture the surface $S$ by removing the basepoints $\{{\bf z}\}$, the remaining periodic domains are in bijection with $H_2(Y - L) \cong Z^{b_2(Y) + {\ell}-1}$, where $\ell$ is the number of components of the link. We say that $D$ is \textit{weakly admissible} if every periodic domain on $S$ has both positive and negative local multiplicities, and require that any Heegaard diagram we use to compute link Floer homology have this property.

\medskip

The construction of the link Floer homology $\widehat{\mathit{HFL}}(Y,L)$ uses the symmetric products $\text{Sym}^{g+n-1}(S)$ consisting of all unordered $(g+n-1)$-tuples of points in $S$. This space is the quotient of $(S)^{g+n-1}$ by the action of the symmetric group $S_{g+n-1}$ permuting the factors of $(S)^{g+n -1}$, and its holomorphic structure is defined by insisting that the quotient map $(S)^{g+n-1} \rightarrow \text{Sym}^{g+n-1}(S)$ be holomorphic. In particular, if $j$ is a complex structure on $S$, there is a natural complex structure $\text{Sym}^{g+n -1}(j)$ on the symmetric product. There are two transversely intersecting submanifolds of $\text{Sym}^{g+n -1}(S)$ of especial interest, namely the two totally real embedded tori $\mathbb T_{\boldsymbol \alpha} = \alpha_1 \times\cdots \times \alpha_{g+n -1}$ and $\mathbb T_{\boldsymbol \beta} = \beta_1 \times\cdots\times \beta_{g+n-1}$. The chain complex $\widehat{\mathit{CFL}}(\mathcal D)$ for knot Floer homology is generated by the finite set of intersection points of $\mathbb T_{\boldsymbol \alpha}$ and $\mathbb T_{\boldsymbol \beta}$. More concretely, a generator of $\widehat{\mathit{CFL}}(\mathcal D)$ is a point ${\bf x} = (x_1 \cdots x_{g+n -1}) \in \text{Sym}^{g+n-1}(S)$ such that each $\alpha$ or $\beta$ curve contains a single $x_i$.

\medskip

In its original form, link Floer homology is computed as follows: let ${\bf x, y}$ be two intersection points in $\widehat{\mathit{CFL}}(\mathcal D)$. Denote by $\pi_2(\bf{x,y})$ the set of Whitney disks $\phi:B_1(0) \rightarrow \text{Sym}^{g+n-1}(S)$ from the unit disk in the complex plane to our symmetric product such that $\phi(-i) = {\bf x}$, $\phi(i) = {\bf y}$ and $\phi$ maps the portion of the boundary of the unit disk with positive real part into $\mathbb T_{\boldsymbol \alpha}$ and the portion with negative real part into $\mathbb T_{\boldsymbol \beta}$. The most common method of studying such maps $\phi$ is to use the following familiar construction of Ozsv{\'a}th and Szab{\'o} to associate to any homotopy class of Whitney disks in $\pi(x,y)$ a domain in $S$. There is a $(g+n-1)$-fold branched cover 
%
\[
S \times \text{Sym}^{g+n -2}(S) \rightarrow \text{Sym}^{g+n-1}(S )
\]

The pullback of this branched cover along $\phi$ is a $(g+n -1)$-fold branched cover of $B_1(0)$ which we shall denote $\Sigma(B_1(0))$. Consider the induced map on $\Sigma(B_1(0))$ formed by projecting the total space of this fibration to $S$.
%
$$
\xymatrix{
\Sigma(B_1(0)) \ar[r] \ar[d] & S \times \text{Sym}^{g+n-2}(S) \ar[r] \ar[d] & S \\
B_1(0) \ar[r]^-{\phi} & \text{Sym}^{g+n-1}(S)
}
$$

We associate to $\phi$ the image of this projection counted with multiplicities; to wit, we let $D = \Sigma a_i D_i$ where $D_i$ are the closures of the components of $S - \cup \alpha_i - \cup \beta_i$ and $a_i$ is the algebraic multiplicity of the intersection of the holomorphic submanifold $V_{x_i} = \{x_i\} \times \text{Sym}^{g+n-2}(S)$ with $\phi(B_1(0))$ for any interior point $x_i$ of $D_i$. The boundary of $D$ consists of $\alpha$ arcs from points of ${\bf x}$ to points of ${\bf y}$ and $\beta$ arcs from points of ${\bf y}$ to points of ${\bf x}$. If $D_i$ contains a basepoint $z_j$, then we introduce some additional notation by letting $a_i = n_{z_i}(\phi)$ be the algebraic intersection number of $z_i \times \text{Sym}^{g+n-2}(S)$ with the image of $\phi$.

\medskip

Given $\phi \in \pi_2({\bf x}, {\bf x})$, we define the \textit{Maslov index} as follows. Recall that $\phi: D^2 \rightarrow \text{Sym}^{g+n-1}(S)$ maps the portion of the boundary of the unit disk $D^2$ in the right half of the complex plane $\mathbb C = \{u+iv: u,v \in \mathbb R\}$ to a loop in $\mathbb T_{\boldsymbol \alpha}$ and the portion of the boundary in the left half to $\mathbb T_{\boldsymbol \beta}$.  Choose a constant trivialization of the orientable real vector bundle $\phi^*(T(\mathbb T_{\boldsymbol \alpha}))$ over $\partial D|_{v\geq 0}$.  We may tensor this real trivialization with $\mathbb C$ and extend to a complex trivialization of $\phi^*(T(\text{Sym}^{g+n-1}(S))$ by pushing across the disk linearly. Relative to this trivialization, the real bundle $\phi^*(T(\mathbb T_{\boldsymbol \beta}))$ over $\partial D|_{v \geq 0}$ induces a loop of real subspaces of $\mathbb C^{g+n-1} = \phi*(T\text{Sym}^{g+n-1}(S))$.  The winding number of this loop is the Maslov index of the map $\phi$.  Notice that we could also have used $\phi^*(J(T(\mathbb T_{\boldsymbol \alpha})))$ and $\phi^*(T(\mathbb T_{\boldsymbol \beta})$, where $J$ is the complex structure on the vector bundle $T\text{Sym}^{g+n-1}(S)$, and obtained the same number.

\medskip

The Maslov index $\mu(\phi)$ can equivalently be computed using the associated domain $\Sigma a_i D_i$ in a formula of Lipshitz's \cite[Proposition 4.2]{MR2240908}. For each domain $D_i$, let $e(D_i)$ be the Euler measure of $D_i$. In particular, if $D_i$ has $2k$ corners, $e(D_i) = 1- \frac{k}{2}$. Let $p_{{\bf x}}(D)$ be the sum of the average of the multiplicities of $D$ at the four corners of each point in ${\bf x}$ and likewise for $p_{{\bf y}}(D)$. Then the Maslov index is
%
\begin{equation} \label{Maslov index formula}
\mu(\phi) = \sum a_i e(D_i) + p_{{\bf x}}(D) + p_{{\bf y}}(D).
\end{equation}

In the case that $[\phi] \in \pi_2({\bf x}, {\bf x})$ is a domain from ${\bf x}$ to itself, and therefore a periodic domain, we have the following alternate interpretation of the Maslov index. Because $\phi(i) = \phi(-i)$, we see that $\phi$ sends $\partial D|_{v \geq 0}$ to a loop in $\mathbb T_{\boldsymbol \alpha}$ and $\partial D_{v \leq 0}$ to a loop in $\mathbb T_{\boldsymbol \beta}$. Therefore we may replace $\phi$ by a map $\widehat{\phi}: S^2 \times I \rightarrow \text{Sym}^{g+n-1}(S)$ which maps $S^1 \times \{1\}$ to $\mathbb T_{\boldsymbol \alpha}$ and $S^1 \times \{0\}$ to $\mathbb T_{\boldsymbol \beta}$.  We then consider the complex pullback bundle $E = \widehat{\phi}^*(T(\text{Sym}^{g+n-1}(S))$ to $S^2 \times I$ and the totally real subbundles $\widehat{\phi}|_{S^1 \times \{1\}}^{*} (T(\mathbb T_{\boldsymbol \alpha}))$ of $E_{S^1 \times \{1\}}$ and  $\widehat{\phi}|_{S^1 \times \{0\}}^{*} (T(\mathbb T_{\boldsymbol \beta}))$ of $E|_{S^1 \times \{0\}}$. The Maslov index is still calculated by trivializing $\widehat{\phi}|_{S^1 \times \{1\}}^{*} (T(\mathbb T_{\boldsymbol \alpha}))$, complexifying, and computing the winding number of the loop of real-half dimensional subspaces in $\mathbb C^{g+n-1}$ represented by $\widehat{\phi}|_{S^1 \times \{0\}}^{*} (T(\mathbb T_{\boldsymbol \beta}))$ with respect to the trivialization. This number classifies the bundle in the following way: complex vector bundles over the annulus whose restriction to the boundary of the annulus carries a canonical real subbundle are in bijection with maps $[(S^1 \times I, \partial(S^1 \times I)), (BU,BO)] = \langle (S^1 \times I, \partial(S^1 \times I)), (BU,BO) \rangle \cong Z$, where the map to $\mathbb Z$ is the Maslov index $\mu(\phi)=\mu(\widehat{\phi})$ \cite[Theorem C.3.7]{MR2045629}.

\medskip

Now let us look at the bundle $E$ over $S^1 \times I$ and its real subbundles over the boundary components of $S^1 \times I$ from a slightly different perspective. The real bundles $\widehat{\phi}|_{S^1 \times \{1\}}^{*} (T(\mathbb T_{\boldsymbol \alpha}))$ and  $\widehat{\phi}|_{S^1 \times \{0\}}^{*} (T(\mathbb T_{\boldsymbol \beta}))$ are orientable, hence trivializable over the circle, so we may choose real trivializations and tensor with $\mathbb C$ to obtain a complex trivialization of $E|_{\partial(S^1 \times I)}$.  We can now regard $E$ as a relative vector bundle $E_{\text{rel}}$ over $(S^1 \times I, \partial(S^1 \times I))$, and consider its relative first Chern class $c_1(E|_{\text{rel}})$.  Equivalently, we may use this trivialization to construct a vector bundle $\widetilde{E}$ over $(S^1 \times I)/ \partial(S^1 \times I) \cong S^2$ such that the pullback $q^*(\widetilde{E})$ along the quotient map is $E$. Then $c_1(\widetilde{E})$ is the relative first Chern class $c_1(E|_{\text{rel}})$ under the identification $H^1(S^2) \simeq H^1(S^1\times I, \partial(S^1 \times I))$. Moreover, isomorphism classes of vector bundles over $S^2$ are in bijection with homotopy classes of maps $[S^2, BU] \cong \langle S^2, BU \rangle = \pi_2(BU) \cong \mathbb Z$, where the identification with $\mathbb Z$ is via the first Chern class.  Using the homotopy long exact sequence of the pair $(BU,BO)$, we observe the following relationship between $\mu$ and $c_1$.
%
\[
\xymatrix{
\pi_2(BU) \ar[r] \ar[d]^{c_1}& \pi_2(BU, BO) \ar[r] \ar[d]^{\mu} & \pi_1(BO) \ar[d] \\
\mathbb Z \ar[r]^-{\times 2} & \mathbb Z \ar[r] & \mathbb Z_2
}
\]
%
Therefore the Maslov index $\mu(\phi)$ is twice the relative first Chern class $c_1(E|_{\text{rel}})$. 

\medskip

The differential $\partial$ on $\widehat{\mathit{CFL}}(\mathcal D)$ counts the dimension of the moduli spaces of pseudo-holomorphic curves of Maslov index one in $\pi_2(\bf{x, y})$.
%
\begin{align*}
\partial({\bf x}) = \sum_{{\bf y} \in \mathbb T_{\boldsymbol \alpha} \cap \mathbb T_{\boldsymbol \beta}} \sum_{\substack{\phi \in \pi_2({\bf x, y}) : \\
	 \mu(\phi) =1 \\
	  n_{w_i}(\phi) = 0 \\
	  n_{z_j}(\phi) = 0 
	  }}
	  \#\left(\frac {M(\phi)}{\mathbb R}\right) {\bf y}
\end{align*}
%
\noindent Ozsv{\'a}th and Szab{\'o} have shown \cite{MR2065507} that this is a well-defined differential. Indeed, once we show that the homology of $\widehat{\mathit{CFL}}(\mathcal D)$ with respect to $\partial$ can be seen as a Floer cohomology theory, this will be a special case of the well-definedness of the differential of Definition \ref{FloerCohomologyDefn}.

\medskip

The complex $\widehat{\mathit{CFL}}(\mathcal D)$ carries a (relative, for our purposes) homological grading called the Maslov grading $M({\bf x})$ which takes values in $\mathbb Z$.  Suppose ${\bf x}$ and ${\bf y}$ are connected by a Whitney disk $\phi$. Then the relative Maslov grading is determined by
%
\begin{align*}
M({\bf x}) - M({\bf y}) &= \mu(\phi) - 2\sum_i n_{w_i}(\phi).
\end{align*}
%

The complex also carries an additional Alexander multigrading ${\bf A} = (A_1,\cdots,A_{\ell})$. This multigrading takes values in an affine lattice $\mathbb H$ over  $H_1(S^3-L; \mathbb Z)\cong H_1(L)$.  Recall that $H_1(S^3-L; \mathbb Z) \cong \mathbb Z^{\ell}$ generated by the homology classes of meridians $\mu_j$ of the component knots $K_j$ of $L$.  Define the lattice $\mathbb H$ to consist of elements
%
%
%
\begin{align*}
\sum_{i=1}^{\ell} a_i[\mu_i]
\end{align*}
%
\noindent where $a_i \in \mathbb Q$ satisfies the property that $2a_i + \ell k(K_i, L-K_i)$ is an even integer. To determine the relative Alexander multigrading, recall that the basepoints $w_{k_{j-1} +1},\cdots,w_{k_j},z_{k_{j-1} +1},\cdots,z_{k_j}$ lie on $K_j$ (with the convention that $k_0 = 0$). Then once again if $\phi$ is a Whitney disk connecting ${\bf x}$ and ${\bf y}$,
%
\begin{align*}
A_j({\bf x}) - A_j({\bf y}) &= \sum_{i=k_{j-1} +1}^{k_j} n_{z_i}(\phi) - \sum_{i=k_{j-1}+1}^{k_j} n_{w_i}(\phi).
\end{align*}
% 
We can also see the relative Alexander multigrading geometrically.  Let ${\bf x},{\bf y} \in \mathbb T_{\boldsymbol \alpha} \cap \mathbb T_{\boldsymbol \beta}$, we find paths
%
\begin{align*}
a:[0,1]\rightarrow \mathbb T_{\boldsymbol \alpha}  \text{  and  }  b:[0,1]\rightarrow \mathbb T_{\boldsymbol \beta}
\end{align*}
%
\noindent such that $\partial a = \partial b = {\bf x} - {\bf y}$.  (For example, $a \cup b$ may be the boundary of a Whitney disk $\phi$ from ${\bf y}$ to ${\bf x}$.)  View these paths as as one-chains on $S \backslash \{{\bf w, z}\}$. Since attaching one- and two-handles to the $\alpha$ and $\beta$ curves on $S$ and filling in three-balls at the basepoints yields $Y$, we obtain a trivial one-cycle in $Y$.  Indeed, a domain $D$ on $\mathcal D$ is the shadow of a Whitney disk if and only if its boundary, viewed as a cycle on $S$, descends to a trivial cycle on $Y$.  However, if we attach $\alpha$ and $\beta$ circles to $S \backslash \{{\bf w,z}\}$ (and no three balls) we obtain the manifold $Y-L$ and a one cycle $\epsilon({\bf x}, {\bf y})$ in $Y-L$. 
%
\begin{align*}
\underline{\epsilon}_{{\bf w,z}}: (\mathbb T_{\boldsymbol \alpha} \cap \mathbb T_{\boldsymbol \beta}) \times (\mathbb T_{\boldsymbol \alpha} \cap \mathbb T_{\boldsymbol \beta}) \rightarrow H_1(Y-L; \mathbb Z)
\end{align*}
%
We obtain the following lemma (which has only been very slightly adjusted from the original to account for the possibility of multiple pairs of basepoints on a link component).

\begin{lemma} \cite[Lemma 3.10]{MR2443092} An oriented $\ell$-component link $L$ in $Y$ induces a map
%
\begin{align*}
\prod H_1(Y-L) \rightarrow \mathbb Z^{\ell}\\
\end{align*}
%
where $\prod_i(\gamma)$ is the linking number of $\gamma$ with the $i$th component $K_i$ of $L$.  In particular, for ${\bf x},{\bf y} \in \mathbb T_{\boldsymbol \alpha}\cap \mathbb T_{\boldsymbol \beta}$ and $\phi \in \pi_2({\bf x}, {\bf y})$, we have
%
\begin{align*}
\prod_i(\underline{\epsilon}_{{\bf w,z}}({\bf x},{\bf y})) = \sum_{n_{i-1}+1}^{n_i} n_{z_j}(\phi) - \sum_{n_{i-1}+1}^{n_i} n_{z_j}(\phi).
\end{align*}


\end{lemma}

\begin{proof}
The proof is nearly identical to the original: $\phi$ induces a nulhomology of $\underline{\epsilon}({\bf x}, {\bf y})$, which meets the $i$th component $K_i$ of $L$ with intersection number $\sum_{n_{i-1}}^{n_1} n_{z_i}(\phi) - \sum_{n_{i-1}}^{n_1} n_{w_i}(\phi)$.
\end{proof}



%
\noindent Formulas for the absolute Maslov grading may be found in \cite[Theorem 3.3]{MR2249248} , and the absolute Alexander multigrading are laid out in \cite[Section 4.2 and Section 8.1]{MR2443092}, but neither will be needed here.

\medskip

The differential $\partial$ lowers the Maslov grading by one and preserves the Alexander multigrading. Therefore $\widehat{\mathit{CFL}}(\mathcal D)$ also splits along Alexander grading. 

\medskip

The homology of $\widehat{\mathit{CFL}}(\mathcal D)$ with respect to the differential $\partial$ is very nearly the link Floer homology of $(Y,K)$. There is, however, a slight subtlety having to do with the number of pairs of basepoints $z_i$ and $w_i$ on $\mathcal D$. Let $V_i$ be a vector space over $\mathbb F_2$ with generators in gradings $(M,(A_1,\cdots,A_{\ell})) = (0,(0,\cdots,0))$ and $(M,(A_1,\cdots,A_j,\cdots,A_{\ell})) = (-1,(0,\cdots,-1,\cdots,0))$, with the $-1$ in the $j$th component.  As before, let $K_j$ carry $n_j$ pairs of basepoints.


\begin{definition}
The homology of the complex $\widehat{\mathit{CFL}}(\mathcal D)$ with respect to the differential $\partial$ is
%
\[
\widetilde{\mathit{HFL}}(\mathcal D) = \widehat{\mathit{HFL}}((S^3,L)) \otimes V_1^{\otimes (n_1-1)}\otimes\cdots\otimes V_{\ell}^{\otimes (n_{\ell}-1)}.
\]

\end{definition}

The theory $\widehat{\mathit{HFL}}(S^3,L)$ is symmetric with respect to the Alexander multigrading as follows.  Let $\widehat{\mathit{HFL}}_d(S^3,L, (A_1,...,A_j))$ be the summand of the link Floer homology of $L$ in Alexander multigrading $(A_1,...,A_j)$ and Maslov grading $d$.

\begin{proposition} \cite[Proposition 8.2]{MR2443092} There is an isomorphism 
%
\[
\widehat{\mathit{HFL}}_d(S^3, L, (A_1,...,A_j)) \cong \widehat{\mathit{HFL}}_{d-\sum A_i}(S^3, L, (-A_1,...,-A_j)).
\]

\end{proposition}

In particular, ignoring Maslov gradings, we see that the link Floer homology is symmetric in each of its Alexander gradings.

\medskip

Before moving on, let us consider one addition differential on the complex $\widehat{\mathit{CFL}}(\mathcal D)$.  Suppose that in addition to the disks we counted previously, we also include disks passing over $z$ basepoints on the component $K_i$ of $L$.  In other words, consider the differential $\partial_{K_j}$ defined as follows.
%
%
\begin{align*}
\partial_{K_j}({\bf x}) = \sum_{{\bf y} \in \mathbb T_{\boldsymbol \alpha} \cap \mathbb T_{\boldsymbol \beta}} \sum_{\substack{\phi \in \pi_2({\bf x, y}) : \\
	 \mu(\phi) =1 \\
	  n_{w_i}(\phi) = 0 \\
	  n_{z_j}(\phi) = 0 \text{ if } z_i \notin K_j
	  }}
	  \#\left(\frac {M(\phi)}{\mathbb R}\right) {\bf y}
\end{align*}

This has the effect of discounting the contribution of the component $K_j$ to the link Floer homology, but of maintaining the effect of an extra $n_j$ pairs of basepoints on the Heegaard surface.  Ergo we have the following proposition.  Let $W$ be a two-dimensional vector space over $\mathbb F_2$ with summands in gradings $(M, (A_1,\cdots,\widehat{A_j},\cdots,A_{\ell})) = (0, (0,...,0))$ and $(M, (A_1,...\widehat{A_j},...A_{\ell})) = (-1, (0,...,0))$. 


\begin{proposition} \label{Ignore Component Proposition} \cite[Proposition 7.2]{MR2443092} The homology of the complex $\widehat{\mathit{CFL}}(S^3, L)$ with respect to the differential $\partial_{K_j}$ is isomorphic to $\widehat{\mathit{HFL}}(S^3, L-K_j) \otimes V_1^{\otimes (n_1-1)} \otimes\cdots\otimes W^{\otimes n_j} \otimes\cdots\otimes V_{\ell}^{\otimes (n_{\ell} -1)}$.


\end{proposition}

We may think of Proposition \ref{Ignore Component Proposition} as the assertion that there is a spectral sequence from the $\mathbb Z^{\ell}$ graded theory $\widehat{\mathit{HFL}}(S^3, L)$ to the $\mathbb Z^{\ell-1}$ graded theory $\widehat{\mathit{HFL}}(S^3, L-K_j)$ by computing all differentials that change the $k_j$th entry of the $\mathbb Z^{\ell}$ multigrading.  This spectral sequence comes with an overall shift in relative Alexander gradings, which is computed by considering fillings of relative $\text{spin}^{\text{c}}$ structures on $S^3-L$ to relative $\text{spin}^{\text{c}}$ structures on $S^3 - (L- K_j)$.  The generally slightly complicated formula admits a simple expression in the case of two-component links, which is the only case of interest to this paper.


\begin{lemma} \label{Gradings Shift Lemma} \cite[Lemma 3.13]{MR2443092} Let $L = K_1 \cup K_2$, and $\lambda= \ell k(K_1,K_2)$. Then suppose $\mathcal D$ is a Heegaard diagram for $(S^3,L)$, and ${\bf x} \in \widetilde{\mathit{CFL}}(\mathcal D)$.  If $(A_1({\bf x}), A_2({\bf x})) = (i,j)$ in the complex $\widetilde{\mathit{CFL}}(\mathcal D)$ with differential $\partial$, then in the complex $\widetilde{\mathit{CFK}}(\mathcal D)$ with differential $\partial_{K_2}$, the Alexander grading of ${\bf x}$ is $A_1({\bf x}) = i - \frac{\lambda}{2}$.

\end{lemma}

That is, forgetting one component of a two-component link has the effect of shifting Alexander gradings of the other component downward by $\frac{\lambda}{2}$.  The proof comes from an analysis of filling relative $\text{spin}^{\text{c}}$ structures; the effect of extending a relative $\text{spin}^{\text{c}}$ structure $\mathfrak s$ on $S^3-\nu L$ to $S^3 - \nu(L-K_j)$ is to shift the Chern class $c_1(\mathfrak s)$ by the Poincar\'{e} dual of the homology class of $K_j$ in $S^3 - \nu(L-K_j)$.  For a two-component link this is a shift by the linking number.


\subsection{Link Floer homology, the multivariable Alexander polynomial, and the Thurston norm}

Recall that the \textit{multivariable Alexander polynomial} of an oriented link $L = K_1 \cup \cdots \cup K_{\ell}$ is a polynomial invariant $\Delta_L(t_1,\cdots,t_{\ell})$ with one variable for each component of the link.  While its relationship to the Alexander polynomials of the component knots is in general slightly complicated, in the case of a two-component $L = K_1 \cup K_2$ Murasugi proved the following using Fox calculus.

\begin{lemma} \cite[Proposition 4.1]{MR0292060} \label{Two Component Link Lemma}
Let $L=K_1\cup K_2$ be an oriented two-component link with $\ell k(K_1, K_2) = \lambda$.  If $\Delta_L(t_1,t_2)$ is the multivariable Alexander polynomial of $L$ and $\Delta_{K_1}(t_1)$ is the ordinary Alexander polynomial of $K_1$, then 
%
\begin{align*}
\Delta_L(t_1, 1) = (1+t+t^2+ \cdots +t^{\lambda-1}) \Delta_{K_1}(t)
\end{align*}


\end{lemma}


The Euler characteristic of link Floer homology encodes the multivariable Alexander polynomial of the link as follows.

\begin{proposition} \cite[Theorem 1.3]{MR2443092} If $L$ is an oriented link,
%
\begin{align*}
\sum_{[{\bf x}] \in \widehat{\mathit{HFL}}(S^3,L)} (-1)^{M({\bf x})} t_1^{A_1({\bf x})}\cdots t_{\ell}^{A_{\ell}({\bf x})} = \begin{cases}
	\left(\prod_{i=1}^{\ell} (t_i^{\frac{1}{2}} - t_i^{-\frac{1}{2}})\right) \Delta_L(t_1,\cdots , t_{\ell}) & \ell>1 \\
	\Delta_L(t_1) & \ell=1
\end{cases}
\end{align*}

\end{proposition}

Link Floer homology also categorifies the Thurston seminorm of the link complement.  Let us recall the definition of the Thurston seminorm on a three manifold with boundary.

\begin{definition}[]
Let $\gamma \in H^2(M, \partial M)$.  The Thurston seminorm $x(\gamma)$ is 
%
\begin{align*}
x(\gamma) = \text{sup}\{ -\chi(S) \}
\end{align*}
%
\noindent where $S$ is any embedded surface in $(M, \partial M)$ with $[S] = \gamma$.


\end{definition}

An important special case occurs when $M$ is the complement of a link $L = K_1 \cup \cdots \cup K_{\ell}$, that is, $M = S^3 - \nu K_1 -\cdots-\nu K_{\ell}$.  Then $H^2(M, \partial M) \cong H_1(L)$, and computing the Thurston seminorm of the element of $H^2(M, \partial M)$ corresponding to $[\sum a_iK_i] \in H_1(L)$ is a matter of computing the minimal Euler characteristic of an embedded surface $F$ whose intersection with a meridian $\mu_i$ of $K_i$ is $a_i$ for each $i$.  In particular, $x([K_i])$ is the minimal Euler characteristic of surface $F$ with boundary one longitude of $K_i$ and an arbitrary number of meridians of the components of $L$. (For practical purposes, one may consider taking a Seifert surface $F$ for $K_i$ and puncturing $F$ wherever it intersects some other component of $L$. However, take note that puncturing a minimal Seifert surface for $K_i$ does not necessarily result in a Thurston-norm minimizing surface.) When $L$ is a knot, this determines the minimal Euler characteristic of a Seifert surface for the knot, and thus determines the genus of the knot.  Because $H_1(L) \cong H^1(S^3 - \nu L)$, we commonly refer to the element of $H^2(M, \partial M)$ which spans $K_i$ as the dual to the homology class of meridian $\mu_i$ of $K_i$.

\medskip


Thurston showed \cite{MR823443} that the Thurston seminorm extends to an $\mathbb R$-valued function of $H_2(S^3 - \nu(L)) \cong H_2(S^3,L)$:
%
\[
x_L: H_2(S^3, L; \mathbb R) \rightarrow \mathbb R. 
\]

Link Floer homology yields a related function.  Recall that $\mathbb H \subset H^2(S^3,L; \mathbb R)\cong H_1(L; \mathbb R)$ is the affine lattice of real second cohomology classes $h=\sum A_i [\mu_i]$ for which $\widehat{\mathit{HFL}}(S^3,L, {\bf A})$ is defined. We have
%
\begin{align*}
y: H^1(S^3 - L; \mathbb R) \rightarrow \mathbb R
\end{align*}
%
\noindent which is defined by 
\[
y(\gamma) = \text{max}_{\{\sum A_i [\mu_i] \in \mathbb H \subset H_1(L; R): \widehat{\mathit{HFL}}(L,{\bf A}) \neq 0\}} |\langle \sum A_i\widehat{[\mu_i]},\gamma \rangle|.
\]

The categorification considers the case of links with no \textit{trivial components}, that is, unknotted components unlinked with the rest of the link.

\begin{proposition} \cite[Thm 1.1]{MR2393424} \label{Statement of Categorification} Let $L$ be an oriented link with no trivial components.  Given $\gamma \in H^1(S^3 - L; \mathbb R)$, the link Floer homology groups determine the Thurston norm of $L$ via the relationship
%
\begin{align*}
x_L(\text{PD}[\gamma]) + \sum_{i=1}^{\ell} | \langle \gamma, \mu_i \rangle | = 2y(\gamma).
\end{align*}

Here $\mu_i$ is the homology class of the meridian for the $i$th component of $L$ in $H_1(S^3 -L; \mathbb R)$, and therefore $| \langle h, \mu_i \rangle |$ is the absolute value of the Kronecker pairing of $h$ with $\mu_i$.


\end{proposition}

We will primarily evaluate this equality on the dual classes to the meridians $\mu_i$ themselves.  Through a slight but traditional abuse of notation, we will sometimes say $x_L([K_i])$ for the Thurston norm of the dual to $\mu_i$. In the case that $K$ is a knot, so that $x_K([K]) = 2g(K) - 1$, Proposition \ref{Statement of Categorification} reduces to the familiar theorem of  \cite[Theorem 1.2]{MR2023281} that the top Alexander grading $i$ for which $\widehat{\mathit{HFK}}(S^3,K,i)$ is nontrivial is the genus of the knot.  In general, observe that if we evaluate on $\mu_i$, we obtain $x_L([K_i]) + 1 = 2y(\mu_i)$.  In other words, the total breadth of the $A_i$ Alexander grading in the link Floer homology is the Thurston norm of the dual to $K_i$ plus one.


\medskip

Before leaving the realm of link Floer homology background, we will require one further result concerning the knot Floer homology of fibred knots.

\begin{proposition}\cite[Thm 1.1]{MR2357503}, \cite[Thm 1.4]{MR2450204} \label{Fiberedness Condition Lemma}
Let $K$ be a knot, and $g(K)$ its genus. Then $K$ is fibered if and only if $\widehat{\mathit{HFK}}(S^3, K, g(K)) = \mathbb Z)$.
\end{proposition}

The forward direction (that if $K$ is fibred, then the knot Floer homology in the top nontrivial Alexander grading is $\mathbb Z$) is due to Ozsv{\'a}th and Szab{\'o} \cite[Theorem 1.1]{MR2153455}, whereas the other direction was proved by Ghiggini \cite{MR2450204} in the case $g=1$ and Ni \cite{MR2357503} in the general case.

\medskip

We are now ready to consider the specific case of periodic knots.



\section{Proofs of Murasugi's and Edmonds' Conditions} \label{Periodic Knots Section}

Let $\widetilde{K}\subset S^3$ be an oriented $q$-periodic knot and $K$ its quotient knot. We will begin by constructing a Heegaard diagram for $(S^3, \widetilde{K} \cup U)$ which is preserved by the action of $\mathbb Z_q$ on $(S^3, \widetilde{K} \cup U)$ and whose quotient under this action is a Heegaard diagram for $(S^3,K\cup U)$.

\subsection{Heegaard diagrams for periodic knots}

As in \cite{Hendricks}, it will be necessary to work with Heegaaard diagrams for $(S^3, K \cup U)$ on the sphere $S^2$. Regard $S^3$ as $\mathbb R^3 \cup \{\infty\}$ and arrange $\widetilde{K}$ such that the unknotted axis of periodicity $U$ is the $z$-axis together with the point at infinity.  Then the projection of $\widetilde{K}$ to the $xy$-plane together with the point at infinity is a periodic diagram $\widetilde{E}$ for $\widetilde{K}$.  Taking the quotient of $(S^3, \widetilde{K})$ by the action of $\mathbb Z_q$ and similarly projecting to the $xy$-plane together with the point at infinity produces a quotient diagram $E$ for $K$.

\medskip

Construct a Heegaard diagram for $K  \cup  U$ as follows: Begin with the diagram $E$ on $S^2 = \mathbb R^2 \cup \{\infty\}$.  Place a basepoint $w_0$ at $\infty$ and $z_0$ at $0$; these will be the sole basepoints on $U$. (This is a slight departure from the notation of Section \ref{Heegaard Floer Background Section}; it will be more convenient to have the indexing start at $w_0$ rather than $w_1$ for the diagrams we construct.) Arrange basepoints $z_1,w_1,\cdots ,z_{n_1},w_{n_1}$ on $K$ such that traversing $K$ in the chosen orientation, one passes through the basepoints in that order.  Moreover, we insist that while travelling from $z_i$ to $w_i$ one passes only through undercrossings and travelling from $w_i$ to $z_{i+1}$ or from $w_n$ to $z_1$ one passes only through overcrossings.  In other words, we choose basepoints so as to make $E$ into a bridge diagram for $K$.  Notice that $n_1$ is at most the number of crossings on the diagram $E$, or half the number of crossings on $\widetilde{E}$.  Encircle the portion of the knot running from $z_i$ to $w_i$ with a curve $\alpha_i$, oriented counterclockwise in the complement of $w_0$. Similarly, encircle the portion of the knot running from $w_{i}$ to $z_{i+1}$ (or from $w_{n_1}$ to $z_1$) with a curve $\beta_i$, oriented counterclockwise in the complement of $w_0$. Notice that both $\alpha_i$ and $\beta_i$ run counterclockwise around $w_i$, and moreover for each $i$, $S^2 \backslash \{\alpha_i, \beta_i\}$ has four components: one each containing $z_{i}, w_i$, and $z_{i+1}$, and one containing all other basepoints. This yields a Heegaard diagram $\mathcal D = (S^2, \boldsymbol{\alpha}, \boldsymbol{\beta}, {\bf w}, {\bf z})$ for $(S^3, K \cup U)$.

\medskip

We may now take the branched double cover of $\mathcal D$ over $z_0$ and $w_0$ to produce a Heegaard diagram $\widetilde{\mathcal D}$ for $(S^3, \widetilde{K} \cup U)$ compatible with $\widetilde{E}$.  This diagram has basepoints $w_0$ and $z_0$ for $U$ and basepoints $z_1^1, w_1^1,\cdots,z_{n_1}^1,w_{n_1}^1, z_1^2,\cdots,w_{n_1}^2,\cdots,z_1^q,\cdots,w_{n_1}^q$ arranged in that order along the oriented knot $\widetilde{K}$. Each adjacent pair $z_i^a$ and $w_i^a$ is encircled by $\alpha_i^a$ a lift of $\alpha_i$, and each adjacent pair $w_{i}^a$ and $z_{i+1}^a$ is encircled by $\beta_i^a$ a lift of $\beta_i$.  (Pairs $w_{n_1}^a$ and $z_1^{a+1}$, as well as $w_{n_1}^q$ and $z_1^1$, are encircled by lifts $\beta_{n_1}^a$ of $\beta_{n_1}$.)  This yields a diagram $\widetilde{\mathcal D} = (S^2, \widetilde{\boldsymbol{\alpha}}, \widetilde{\boldsymbol{ \beta}}, \widetilde{\bf w}, \widetilde{\bf z})$ with $qn_1$ each of $\alpha$ and $\beta$ curves and $qn_1 +1$ pairs of basepoints.

\begin{remark}
The notation above is not quite the notation of \cite{Hendricks}, in which the two lifts of a curve $\alpha_i$ in a Heegaard surface $\mathcal D$ to its double branched cover $\widetilde{\mathcal D}$ were $\widetilde{\alpha_i}$ and $\tau(\widetilde{\alpha_i})$.  In the new slightly more streamlined notation, adopted in view of the need to work with $q$-fold branched covers and to consider multiple lifts of some of the basepoints, these two curves would be $\alpha_i^1$ and $\alpha_i^2$.
\end{remark}

Let us pause to introduce some notation on the diagram $\mathcal D$ that will be useful in Section \ref{Stable Normal Triv Section}.  Let $x_i$ be the single positive intersection point between $\alpha_i$ and $\beta_i$ and $y_i$ the negative intersection point.  Moreover, let $F_i$ be the closure of the component of $S - \alpha_i - \beta_i$ containing $z_{i}$ and $E_i$ be the closure of the component of $S - \alpha_i -\beta_i$ containing $z_{i+1}$ (or $z_1$ if $i=n_1$).  Then $P_i = E_i - F_i$ is a periodic domain of index zero on $\mathcal D$.  Finally, let $\gamma_i$ be the union of the arc of $\alpha_i$ running from $x_i$ to $y_i$ and the arc of $\beta_i$ running from $x_i$ to $y_i$.  In particular, this specifies that $\gamma_i$ has no intersection with any $\alpha$ or $\beta$ curves other than $\alpha_i$ and $\beta_i$, and moreover the component of $S - \gamma_i$ which does not contain $w_0$ contains only a single basepoint $w_i$.

\medskip



\begin{figure}
\caption{An equivariant Heegaard diagram $\widetilde{\mathcal D}$ for the trefoil together with the unknotted axis, and its quotient Heegaard diagram $\mathcal D$ for the Hopf link.}
\label{Trefoil Heegaard Diagrams Figure}
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\end{figure}



Our next goal will be to investigate the behavior of the relative Maslov and (particularly) Alexander gradings of generators of $\widehat{\mathit{CFK}}(\mathcal D)$ and $\widehat{\mathit{CFK}}(\widetilde{\mathcal D})$.  We begin with two relatively simple lemmas. As before, let $\tau$ be the involution on $(S^3, \widetilde{K})$ (and on $\widetilde{\mathcal D}$).  Let $\tau^{\#}$ be the induced involution on $\widehat{\mathit{CFK}}(\widetilde{\mathcal D})$.

\begin{lemma}
The induced map $\tau^{\#}$ preserves Alexander and Maslov gradings.
\end{lemma}

\begin{proof}
Let ${\bf s} \in \widehat{\mathit{CFK}}(\widetilde{\mathcal D})$.  Choose a generator ${\bf x} \in \widehat{\mathit{CFK}}(\mathcal D)$, and let $\widetilde{\bf x} = \pi^{-1}(x)$, such that $\widetilde{\bf x}$ is a generator in $\widehat{\mathit{CFK}}(\widetilde{\mathcal D})$ which is invariant under $\tau^{\#}$.  Choose a Whitney disk $\phi$ in $\pi_2({\bf s},\widetilde{\bf x})$ and let  $D$ be its shadow onon $\mathcal D$.  Then $\tau \circ \phi$ is a Whitney disk in $\pi_2(\tau({\bf s}),\widetilde{\bf x})$ with shadow $\tau(D)$. Furthermore, since $w_0$ and $z_0$ are fixed by the involution, $n_{z_0}(D)=n_{z_0}(D)$ and $n_{w_0}(D) = n_{w_0}(\tau(D))$, whereas since the remaining basepoints are interchanged by the involution, we have $\sum_{i=0}^{n_1} n_{z_i}(D) = \sum_{i=0}^{n_1} n_{z_i}(\tau(D))$ and $\sum_{i=0}^{n_1} n_{w_i}(D) = \sum_{i=0}^{n_1} n_{w_i}(\tau(D))$.  These equalities imply that $M(\widetilde{\bf x}) - M({\bf s}) = M(\widetilde{\bf x}) - M(\tau^{\#}({\bf s}))$, and similarly for $A_1$ and $A_2$.  Therefore ${\bf s}$ and $\tau^{\#}{\bf s}$ are in identical gradings.

\end{proof}

\begin{lemma} \label{Maslov Index Lemma}
Let $\phi \in \pi_2({\bf x},{\bf y})$ be a Whitney disk between generators ${\bf x}$ and ${\bf y}$ in $\widehat{\mathit{CFK}}(\mathcal D)$ with Maslov index $\mu(\phi)$, with shadow the domain $D$ on $\mathcal D$. There is a Whitney disk $\widetilde{\phi} \in \pi_2(\widetilde{{\bf x}},\widetilde{{\bf y}})$ with shadow the domain $\pi^{-1}(D)$ on $\widetilde{\mathcal D}$, and $\mu(\widetilde{\phi}) = q \mu(\phi) - (q-1)(n_{z_0}(\phi) + n_{w_0}(\phi))$.

\end{lemma}

\begin{proof}

The boundary of the lift $\pi^{-1}(D)$ is trivial as a one cycle in $S^3$, implying that $\pi^{-1}(D)$ is the shadow of a Whitney disk $\widetilde{\phi} \in \pi_2(\widetilde{{\bf x}},\widetilde{{\bf y}})$.  We will compare the Maslov index of $\phi$ with the Maslov index of $\widetilde{\phi}$ using the formula \ref{Maslov index formula}.  As in that formula, we will write $D$ as a sum of the closures of the components of $S^2 - {\boldsymbol \alpha} - {\boldsymbol \beta}$. Say there are $m$ such components, and label them as follows.  There are two domains in $S^2 - {\boldsymbol \alpha} -{\boldsymbol \beta}$ which contain a branch point.  Let these be $D_1$ containing $z_0$ and $D_2$ containing $w_0$.  Let the shadow of $\phi$ be $a_1D_1 + a_2D_2 +\sum_{i=3}^{m} a_iD_i$. Then the Maslov index of $\phi$ is
%
\[
\mu(\phi) = \sum_i a_i e(D_i) + p_{\bf x}(D) + p_{\bf y}(D)
\]
%
Let us now consider applying the same formula to $\pi^{-1}(D) = \sum_i a_i \pi^{-1}(D_i)$. For $i\geq 3$, the lift $\pi^{-1}(D_j)$ of $D_j$ consists of $q$ copies of $D_i$, and by additivity of the Euler measure we see that $e(\widetilde{D}_i) = qe(D_i)$.  For $i=1,2$, let $2k_i$ be the number of corners of $D_i$.  Then $\pi^{-1}(D_i) = \widetilde{D}_i$ is a single component of $S^2 \ \widetilde{\boldsymbol \alpha} - \widetilde{\boldsymbol \beta}$ with $2qk_i$ corners and Euler measure $e(\widetilde{D}_i) = 1 - \frac{qk_i}{2} = q(e(D_i)) - (q-1)$.  Notice, furthermore, that $p_{\widetilde{\bf x}}(\widetilde{D}) = q p_{\bf x}(D)$ and $p_{\widetilde{\bf y}}(\widetilde{D}) = q p_{\bf y}(D)$. Therefore we compute
%
\begin{align*}
\mu(\widetilde{\phi}) &= \sum_i a_i e(\pi^{-1}(D_i)) + p_{\widetilde{\bf x}}(\widetilde{D}) + p_{\widetilde{\bf y}}(\widetilde{D}) \\
						&= a_1 e(\widetilde{D}_1) + a_2 e(\widetilde{D}_2) + q \sum_{i=3}^m e(D_i) + qp_{\bf x}(D) + q p_{\bf y}(D)\\
						&=a_1 (qe(D_1) - (q-1)) + a_2(qe(D_2) - (q-1)) +  q \sum_{i=3}^m e(D_i) + qp_{\bf x}(D) + q p_{\bf y}(D)\\
						&=q\mu(\phi) - (q-1)(k_1 + k_2).
\end{align*}

Since $k_1+k_2$ is exactly $n_{z_0}(D) + n_{w_0}(D)$ the total algebraic intersection of $\phi$ with the branch points, this proves the result. \end{proof}

We can now construct the relationship between the Alexander gradings of the generators of $\widehat{\mathit{CFL}}(\widetilde{\mathcal D})$ and $\widehat{\mathit{CFL}}(\mathcal D)$.  For the case of a $q$-periodic knot, we will look only at the relative gradings; later in the particular case of a doubly-periodic knot we will fix the absolute gradings using symmetries of link Floer homology.  Let $\pi:\widetilde{\mathcal D} \rightarrow \mathcal D$ be the restriction of the branched covering map $(S^3, \widetilde{K} \cup U) \rightarrow (S^3, K\cup U)$.  We have the following, which is analogous to \cite[Lemma 3.1]{MR2443111}.


\begin{lemma} \label{Generator Matching Lemma}

Let ${\bf s} \in \widehat{\mathit{CFL}}(\widetilde{\mathcal D})$, thought of as an $qn_1$-tuple of points on $S^2$ with one point on each $\alpha_i^{k}$ and one on each $\beta_i^{k}$. Consider its projection $\pi({\bf s})$ to $qn_1$ points on $\mathcal D$. There is a (not at all canonical) way to write $\pi({\bf s})$ as ${\bf s_1} \cup {\bf s_2} \cup \cdots \cup {\bf s_q}$ a union of generators in $\widehat{\mathit{CFL}}(\mathcal D)$.

\end{lemma}

The proof of this lemma (pointed out by Adam Levine) is an application of the following combinatorial result of Hall \cite{Hall}.  Let $A$ be a set, and $\{ A_i\}_{i=1}^m$ be a collection of finite subsets (a version also exists for infinitely many $A_i$). A \textit{system of distinct representatives} is a choice of elements $j_i \in A_i$ for each $i$ such that $j_{i_1} \neq j_{i_2}$ if $i_1 = i_2$.  Hall's theorem gives conditions under which a system of distinct representatives exists.


\begin{theorem} \cite[Theorem 1]{Hall} Let $\{A_i\}_{i=1}^m$ be finitely many subsets of a set $A$.  Then a system of distinct representatives exists if and only if, for any $1\leq s\leq m$ and $1<i_1<...<i_s<m$,  $A_{i_1} \cup ... \cup A_{i_s}$ contains at least $s$ elements.
\end{theorem}

Using this, we may prove Lemma \ref{Generator Matching Lemma}

\begin{proof}[Proof of Lemma \ref{Generator Matching Lemma}]  Let $A = \{1,...,q\}$. For $1 \leq i \leq n_1$, let $A_i$ be a set of integers $j$, with $1\leq j \leq n_1$, such that each $j$ appears once in $A_i$ for every point of $\pi({\bf s}) \cap (\alpha_i \cap \beta_j)$.  That is, the sets $A_i$ record how many intersection points on $\alpha_i$ also lie on $\beta_j$.  Notice that there are $q$ elements in each $A_i$, and each $j$ appears exactly $q$ times in $\coprod_i A_i$.  We claim the sets $A_i$ satisfy the condition of Hall's theorem.  For given $1 \leq i_1 \leq ... \leq i_m \leq n_1$, the disjoint union $\coprod_{k=1}^{m} A_{i_k}$ contains $qm$ elements, and therefore must contain at least $m$ different integers $j$.  Therefore $\bigcup_{k=1}^{m} A_{i_k}$ contains at least $m$ elements.  Hence we can choose a set of distinct representatives $j_i$ in $A_i$.  There is a generator ${\bf s_1}$ consisting of points in $\pi({\bf s})$ on $\alpha_i \cap \beta_{j_i}$.  Remove these points from $\pi({\bf s})$ (and the individual $j_i$ from the sets $A_i$, producing new sets $A_i'$) and repeat the argument, now with $q-1$ elements in each $A_i'$ and $q-1$ appearances of each symbol $j$ in $\coprod_i A_i'$.  After $q$ repetetions, we have broken ${\bf s}$ into ${\bf s_1}\cup...\cup {\bf s_q}$, where each ${\bf s_i}$ is a generator for  $\widehat{\mathit{CFL}}(\mathcal D)$.  This choice of partition is not at all unique.
\end{proof}




We can start by determining the relative Alexander gradings of generators of $\widehat{\mathit{CFL}}(\widetilde{\mathcal D})$ which are invariant under the action of $\mathbb Z_q$ on $\widetilde{\mathcal D}$; that is, exactly those generators which are total lifts of generators in $\widehat{\mathit{CFL}}(\mathcal D)$ under the projection map $\pi$.

\begin{lemma}

Let ${\bf x}, {\bf y} \in \mathbb T_{\boldsymbol \alpha} \cap \mathbb T_{\boldsymbol \beta}$ and $\widetilde{\bf x}, \widetilde{\bf y}$ be their total lifts in $\mathbb T_{\widetilde{\boldsymbol \alpha}} \cap \mathbb T_{\widetilde{\boldsymbol \beta}}$.  Then
%
%
\begin{align*}
A_1(\widetilde{\bf y}) - A_1(\widetilde{\bf x}) &= q\left(A_1({\bf y}) - A_1({\bf x})\right) \\
A_2(\widetilde{\bf y}) - A_2(\widetilde{\bf x}) &= A_2({\bf y}) - A_2({\bf x}).
\end{align*}

\end{lemma}

\begin{proof}
Let $D$ be a domain from ${\bf x}$ to ${\bf y}$ on $\mathcal D$.  Then $\pi^{-1}(D) = \widetilde{D}$ is a domain from ${\bf x}$ to ${\bf y}$ on $\widetilde{\mathcal D}$. Since $z_0$ and $w_0$ are branch points of $\pi$, we see that $n_{z_0}(\widetilde{D}) = n_{z_0}(D)$ and $n_{w_0}(\widetilde{D}) = n_{w_0}(D)$.  Therefore
%
\begin{align*}
A_2(\widetilde{\bf x}) - A_2(\widetilde{\bf y}) &= n_{z_0}(\widetilde{D}) - n_{w_0}(\widetilde{D}) \\
									&=n_{z_0}(D) - n_{w_0}(D) \\
									&=A_2({\bf x}) - A_2({\bf y})
\end{align*}


However, for $i\neq 0$, each of $z_i$ and $w_i$ basepoints on $\mathcal D$ has $q$ preimages in $\widetilde{\mathcal D}$.  Moreover $n_{z_i^j}(\widetilde{D}) = n_{z_i}(D)$ for all $1 \leq i \leq n_1$ and $1 \leq j \leq q$, and similarly for $w_i^j$, so we compute as follows.
%
\begin{align*}
A_1(\widetilde {\bf x}) - A_1(\widetilde{\bf y}) &= \sum_{i=1}^{n_1}\sum_{j=1}^q n_{z_i^j}(\widetilde{D}) - \sum_{i=1}^{n_1}\sum_{j=1}^q n_{w_i^j}(\widetilde{D}) \\
&= q \left( \sum_{i=1}^{n_1} n_{z_i}(D) - \sum_{i=1}^{n_1} n_{w_i}(D)\right) \\
&= q \left( A_1({\bf x}) - A_1({\bf y})\right) \qedhere
\end{align*}

\end{proof}

Finally, we can use this fact to compute the relative Alexander gradings in $\widehat{\mathit{CFL}}(\widetilde{\mathcal D})$.

\begin{lemma}

Let ${\bf s}, {\bf r}$ be generators of $\widehat{\mathit{CFL}}(\widetilde{\mathcal D})$, whose projection to $\mathcal D$ can be written as $\pi_{*}({\bf s}) = {\bf s_1} \cup \cdots \cup {\bf s_q}$ and ${\bf r} = {\bf r_1} \cup \cdots \cup {\bf r_q}$. Then the relative Alexander gradings between ${\bf s}$ and ${\bf r}$ is described by 
%
\begin{align*}
A_1({\bf s}) - A_1({\bf r}) = \sum_{j=1}^{q} \left(A_1({\bf s_j}) - A_1({\bf r_j})\right) \\
A_2({\bf s}) - A_2({\bf r}) = \frac{1}{q} \left(\sum_{j=1}^{q} \left(A_1({\bf s_j}) - A_1({\bf r_j})\right)\right).
\end{align*}
\end{lemma}


\begin{proof}

The proof is quite similar to the first half of the argument of \cite[Proposition 3.4]{MR2443111}. As in Section \ref{Heegaard Floer Background Section}, choose paths $a:[0,1] \rightarrow \mathbb T_{\widetilde{\boldsymbol \alpha}}$, $b:[0,1] \rightarrow \mathbb T_{\widetilde{\boldsymbol \beta}}$, with $\partial a = \partial b = {\bf s} - {\bf r}$. Then let $\epsilon({\bf s},{\bf r}) = a -b $ be a one-cycle in $H_1(S^3 - (\widetilde{K} \cup U); \mathbb Z)$.  Consider the projection $\pi_{*}(\epsilon({\bf s},{\bf r}))$ to $\mathcal D$.  The restriction of this one-cycle to any $\alpha$ or $\beta$ curve consists of $q$ possibly overlapping arcs. By adding copies of the $\alpha$ or $beta$ circle if necessary, we may arrange that these arcs connect a point in ${\bf s_j}$ to a point in ${\bf r_j}$.  That is, modulo $\alpha$ and $\beta$ curves, which have linking number zero with the knot, $\pi_{*}\epsilon({\bf s},{\bf r}) \equiv \epsilon({\bf r_1},{\bf s_1}) \cdots \epsilon({\bf r_q}, {\bf s_q})$.  Notice also that $\ell k\left(\pi_{*}\epsilon({\bf s},{\bf r}), K\right) = \ell k \left(\epsilon({\bf s},{\bf r}), \widetilde{K}\right)$, whereas $\ell k\left(\pi_{*}\epsilon({\bf s},{\bf r}), U\right) = q \ell k \left(\epsilon({\bf s},{\bf r}), U \right)$. Therefore we compute
%
%
\begin{align*}
A_1({\bf r}) - A_1({\bf s}) &= \ell k\left(\epsilon({\bf s},{\bf r}),\widetilde{K}\right) \\
							&=\ell k(\pi_{*}(\epsilon({\bf s},{\bf r})), K) \\
							&=\sum_{j=1}^{q}\ell k \left(\epsilon\left({\bf s_j},{\bf r_j}\right),K \right)\\
							&=\sum A_1({\bf r}) - A_1({\bf s})\\
\end{align*}
%
and moreover
%
\begin{align*}
A_2({\bf r}) - A_2({\bf s}) &= \ell k\left(\epsilon({\bf s},{\bf r}), U\right)\\
							&=\frac{1}{q}\ell k\left(\pi_{*}(\epsilon({\bf s},{\bf r})), U\right)\\
							&=\frac{1}{q}\left(\sum_{j=1}^{q}\left(\ell k(\epsilon({\bf s_j},{\bf r_j}),U 											\right)\right) \\
							&=\frac{1}{q}\left(\sum_{j=1}^{q}\left( A_1({\bf r_j}) - A_1({\bf s_j})\right)\right)\qedhere
\end{align*}
\end{proof}








We may now give a proof of Theorem \ref{Murasugi Theorem}.

\begin{proof}[Proof of Theorem \ref{Murasugi Theorem}]

Let $\widetilde{K}$ be a periodic knot with period $q=p^r$ for some prime $p$, and $K$ be its quotient knot, and let $\lambda = \ell k(\widetilde{K},U) = \ell k(K,U)$.  Choose a periodic diagram $\widetilde{D}$ for $(S^3, \widetilde{K} \cup U)$ and its quotient diagram $D$ for $(S^3, K \cup U)$ as outlined above.  

\medskip

Consider the Euler characteristic of $\widehat{\mathit{CFL}}(\widetilde{\mathcal D})$ computed modulo $p$.  Let ${\bf s}$ be a generator in  $\widehat{\mathit{CFL}}(\widetilde{\mathcal D})$. Either ${\bf s} = \widetilde{{\bf y}} = \pi^{-1}(\bf y)$ for some ${\bf y} \in \widehat{\mathit{CFL}}(\mathcal D)$, and thus ${\bf s}$ is invariant under the action of $\tau$, or the order of the orbit of ${\bf s}$ under the action of $\tau$ is a multiple of $p$.  Since the action preserves the Alexander and Maslov gradings, modulo $p$ the terms of the Euler characteristic of $\widehat{\mathit{CFL}}(\widetilde{\mathcal D})$ corresponding to noninvariant generators sum to zero.  Moreover, there is a one-to-one correspondence between generators ${\bf y}$ of $\widehat{\mathit{CFL}}(\mathcal D)$ and their total lifts $\pi^{-1}({\bf y}) = {\bf s}$ in $\widehat{\mathit{CFL}}(\widetilde{\mathcal D})$.

\medskip

This correspondence preserves relative Alexander $A_2$-gradings, and multiplies Alexander $A_1$-gradings by a factor of $q$.  We also claim that it preserves the parity of relative Maslov gradings if $p$ is odd.  In particular, any two generators ${\bf x}$ and ${\bf y}$ of $\widehat{\mathit{CFL}({\mathcal D})}$ are joined by a domain $D$ which does not pass over $w_0$. Let $\widetilde{ D}$ be the lift of this domain. Then the Maslov index $\mu(\widetilde{D})$ is equal to $q\mu(D) - (q-1)n_{z_0}(D)$ by Lemma \ref{Maslov Index Lemma}, and we have
%
\begin{align*}
M(\widetilde{{\bf x}}) - M(\widetilde{{\bf y}}) &= \mu(\widetilde{D}) - 2\sum_{i=1}^{n} \left(n_{w_i^1}(\widetilde{D}) + \cdots + n_{w_i^q}(\widetilde{D})\right) \\
&= q\mu(D) -(q-1) n_{z_0}(D) - 2q\sum_{i=1}^n n_{w_i}(D) \\
&\equiv \mu(D) - 2\sum_{i=1}^n n_{w_i}(D) \text{ mod } 2\\
&\equiv M({\bf x}) - M({\bf y}) \text{ mod } 2
\end{align*}

Here we have used the assumption that $p$, and therefore $q$, is odd, and $q-1$ is even.  Ergo if $p$ is odd, $(-1)^{M(\widetilde{{\bf x}})} = \pm (-1)^{M({\bf x})}$ where the choice of sign is the same for all generators ${\bf x}$ of $\widehat{\mathit{CFL}}(\mathcal D)$.  (If $p=2$ the sign is of course immaterial.)

\medskip

Therefore, we capture the following equality.
%
\begin{align*}
\chi(\widehat{\mathit{CFL}}(\widetilde{\mathcal D})) (t_1, t_2) \doteq \chi(\widehat{\mathit{CFL}}(\widetilde{\mathcal D}) )(t_1^q, t_2) \text{ mod } p
\end{align*}

\noindent Here $\doteq$ denotes equivalence up to an overall factor of $\pm t_1^{i_1}t_2^{i_2}$.

\medskip

However, $\chi(\widehat{\mathit{CFL}}(\widetilde{\mathcal D}))$ is $\chi(\widetilde{\mathit{HFL}}(\widetilde{\mathcal D}))$ and similarly $\chi(\widehat{\mathit{CFL}}(\mathcal D))$ is $\chi(\widetilde{\mathit{HFL}}(\mathcal D))$.  Hence we may express the Euler characteristics of the chain complexes as the multivariable Alexander polynomials of $\widetilde{L} = \widetilde{K} \cup U$ and $L = K \cup U$ multiplied by appropriate powers of $(1- t_1^{-1})$ and $(1-t_2^{-1})$ according to the number of basepoints on each component of each link. Therefore the equality above reduces to 
%
\begin{align*}
\Delta_{\widetilde{L}}(t_1, t_2)(1-t_1^{-1})^{qn}(1-t_2^{-1}) &\doteq \Delta_{L}(t_1^q, t_2)(1-(t_1^{-1})^{q})^n(1-t_2^{-1}) \text{ mod } p\\
\Delta_{\widetilde{L}}(t_1, t_2)(1-t_1^{-1})^{qn} &\doteq \Delta_{L}(t_1^q, t_2)(1-(t_1^{-1})^{q})^n \text{ mod } p.
\end{align*}
%
\noindent Recalling that $q$ is a power of $p$, and that therefore $(a+b)^q \equiv a^q + b^q$ mod $p$, we may reduce farther.
%
\begin{align*}
\Delta_{\widetilde{L}}(t_1,t_2)(1-t_1^{-1})^{qn} &\doteq \Delta_{L}(t_1^q, t_2)(1-t_1^{-1})^{qn} \text{ mod } p\\
\Delta_{\widetilde{L}}(t_1,t_2) &\doteq \Delta_{L}(t_1^q, t_2) \text{ mod } p
\end{align*}
%
\noindent We now set $t_2 = 1$.  By Lemma \ref{Two Component Link Lemma}, this reduces the equality above to
%
\begin{align*}
\Delta_{\widetilde{K}}(t_1)(1 + t_1 + \cdots + t_1^{\lambda-1}) &\doteq \Delta_{K}(t_1^q)(1 + t_1^q + (t_1^q)^2 + \cdots + ((t_1)^q)^{\lambda-1} \text{ mod } p.
\end{align*}
%
\noindent Again using the fact that $q$ is a power of $p$, we produce
%
\begin{align*}
\Delta_{\widetilde{K}}(t_1)(1 + t_1 + \cdots + t_1^{\lambda-1}) &\doteq (\Delta_{K}(t_1))^q(1 + t_1 + t_1^2 + \cdots + t_1^{(\lambda-1)})^q \text{ mod } p \\
\Delta_{\widetilde{K}}(t_1) &\doteq (\Delta_K(t_1))^q (1 + t + \cdots + t^{\lambda -1})^{q-1} \text{ mod } p.
\end{align*}
%
\noindent This last is Murasugi's condition.\end{proof}

\subsection{Spectral sequences for doubly-periodic knots}

From now on we restrict ourselves entirely to the case of $\widetilde{K}$ a doubly-periodic knot.  Moreover, we insist that $\widetilde{K}$ be oriented such that $\ell k(\widetilde{K},U) = \lambda$ is positive.  Notice also that $\lambda$ is necessarily odd; otherwise $\tilde{K}$ would be disconnected. We proceed to explain how Corollary \ref{Genus Inequality Corollary} follows from Theorem \ref{Knot Floer Homology Spectral Sequence}.  Consider the map $\tau^{\#}: \widehat{\mathit{CFL}}(\widetilde{\mathcal D}) \rightarrow \widehat{\mathit{CFL}}(\widetilde{\mathcal D})$ induced by the involution $\tau$ on $\widetilde{\mathcal D}$.  As a consequence of our application of Seidel and Smith's localization theory to the symmetric products $\text{Sym}^{2n_1}(S^2 \backslash \{{\bf \widetilde{w},\widetilde{z}}\})$ and $\text{Sym}^{n_1}(S^2 \backslash \{{\bf w,z}\})$, we will see that $\tau^{\#}$ is a chain map for this Heegaard diagram. The spectral sequence of Theorem \ref{Link Floer Homology Spectral Sequence} is induced by the double complex
%
$$
\xymatrix{
 & \ar[d]^-{\partial} & \ar[d]^-{\partial} & \ar[d]^-{\partial} & \\
0 \ar[r] & \widehat{\mathit{CFL}}_{i+1}(\widetilde{\mathcal D}) \ar[r]^-{1 + \tau^{\#}} \ar[d]^-{\partial} & \widehat{\mathit{CFL}}_{i+1}(\widetilde{\mathcal D}) \ar[r]^-{1 + \tau^{\#}} \ar[d]^-{\partial}& \widehat{\mathit{CFL}}_{i+1}(\widetilde{\mathcal D}) \ar[r]^-{1 + \tau^{\#}} \ar[d]^-{\partial} &\\
0 \ar[r] & \widehat{\mathit{CFL}}_{i}(\widetilde{\mathcal D}) \ar[r]^-{1 + \tau^{\#}} \ar[d]^-{\partial} & \widehat{\mathit{CFL}}_{i}(\widetilde{\mathcal D}) \ar[r]^-{1 + \tau^{\#}} \ar[d]^-{\partial}& \widehat{\mathit{CFL}}_{i}(\widetilde{\mathcal D}) \ar[r]^-{1 + \tau^{\#}} \ar[d]^-{\partial} &\\
0 \ar[r] & \widehat{\mathit{CFL}}_{i-1}(\widetilde{\mathcal D}) \ar[r]^-{1 + \tau^{\#}} \ar[d]^-{\partial} & \widehat{\mathit{CFL}}_{i-1}(\widetilde{\mathcal D}) \ar[r]^-{1 + \tau^{\#}} \ar[d]^-{\partial}& \widehat{\mathit{CFL}}_{i-1}(\widetilde{\mathcal D}) \ar[r]^-{1 + \tau^{\#}} \ar[d]^-{\partial}& \\
 & & & &
}
$$

\begin{definition}
The homology of the complex $(\widehat{\mathit{CFL}}(\widetilde{\mathcal D} \otimes \mathbb Z_2[[\theta]], d + \theta(1+ \tau^{\#}))$ is $\widehat{\mathit{HFL}}_{\text{borel}}(\widetilde{\mathcal D})$.
\end{definition}

Computing vertical differentials first, the $E^1$ page of this spectral sequence is $\widetilde{\mathit{HFL}}(\widetilde{\mathcal D}) \otimes \mathbb Z_2[[\theta]]$; we will see that after tensoring with $\theta^{-1}$, the $E^{\infty}$ page is $\mathbb Z((\theta))$-isomorphic to $\widetilde{\mathit{HFL}}(\mathcal D) \otimes \mathbb Z_2((\theta))$ under a map which preserves both Alexander gradings.  Since  $\partial$ and $\tau^{\#}$ also preserve both Alexander gradings on $\widehat{\mathit{CFL}}(\widetilde{D})$, the spectral sequence splits along the grading $(A_1,A_2)$.  

\medskip





The knot Floer homology spectral sequence of Theorem \ref{Knot Floer Homology Spectral Sequence} arises from a similar double complex.  Recall that for a link $L=K_1\cup\cdots\cup K_\ell$, the differential $\partial_{K_j}$ corresponds to forgetting the component $K_j$ of the link.  Therefore for the link $L=\widetilde{K}\cup U$, let $\partial_U$ be the differential corresponding to forgetting the component $U$ of $L$.
%
$$
\xymatrix{
 & \ar[d]^-{\partial_U} & \ar[d]^-{\partial_U} & \ar[d]^-{\partial_U} & \\
0 \ar[r] & \widehat{\mathit{CFL}}_{i+1}(\widetilde{\mathcal D}) \ar[r]^-{1 + \tau^{\#}} \ar[d]^-{\partial_U} & \widehat{\mathit{CFL}}_{i+1}(\widetilde{\mathcal D}) \ar[r]^-{1 + \tau^{\#}} \ar[d]^-{\partial_U}& \widehat{\mathit{CFL}}_{i+1}(\widetilde{\mathcal D}) \ar[r]^-{1 + \tau^{\#}} \ar[d]^-{\partial_U} &\\
0 \ar[r] & \widehat{\mathit{CFL}}_{i}(\widetilde{\mathcal D}) \ar[r]^-{1 + \tau^{\#}} \ar[d]^-{\partial_U} & \widehat{\mathit{CFL}}_{i}(\widetilde{\mathcal D}) \ar[r]^-{1 + \tau^{\#}} \ar[d]^-{\partial_U}& \widehat{\mathit{CFL}}_{i}(\widetilde{\mathcal D}) \ar[r]^-{1 + \tau^{\#}} \ar[d]^-{\partial_U} &\\
0 \ar[r] & \widehat{\mathit{CFL}}_{i-1}(\widetilde{\mathcal D}) \ar[r]^-{1 + \tau^{\#}} \ar[d]^-{\partial_U} & \widehat{\mathit{CFL}}_{i-1}(\widetilde{\mathcal D}) \ar[r]^-{1 + \tau^{\#}} \ar[d]^-{\partial_U}& \widehat{\mathit{CFL}}_{i-1}(\widetilde{\mathcal D}) \ar[r]^-{1 + \tau^{\#}} \ar[d]^-{\partial_U}& \\
 & & & &
}
$$


\begin{definition}
The homology of the complex $(\widehat{\mathit{CFK}}(\mathcal D \otimes \mathbb Z_2[[\theta]], d_U + \theta(1+ \tau^{\#}))$ is $\widehat{\mathit{HFL}}_{\text{borel}}(\mathcal D)$.
\end{definition}

As $\partial_U$ does not preserve Alexander $A_2$ gradings relative to the axis $U$, neither does the spectral sequence; however, it still splits along Alexander $A_1$ gradings, the grading relative to the knot itself.

\medskip

Let us consider how we might fix the relationship between the absolute Alexander gradings of $\widetilde{\mathit{HFL}}(\widetilde{\mathcal D})$ and $\widetilde{\mathit{HFL}}(\mathcal D)$.  Let $x_L([K])$ be the Thurston seminorm of the class dual to $[K]$ in $H^2(S^3 - \nu(L), \partial(S^3 - \nu(L)))$ and $x_{\widetilde{L}}([\widetilde{K}])$ be the Thurston seminorm of the class dual to $[\widetilde{K}]$ in $H^2(S^3 - \nu(\widetilde{L}), \partial(S^3 - \nu(\widetilde{L})))$.  Notice that if $F$ is a Thurston-norm minimizing surface for the class dual to $[K]$ of Euler characteristic $\chi(F) = -x_L([K])$, then the preimage $\widetilde{F} = \pi^{-1}(F)$ under the ordinary double cover $\pi: S^3-\nu(\widetilde{L}) \rightarrow S^3-\nu(L)$ is an embedded surface representing the class dual to $[\widetilde{K}]$ in $H^2(S^3 - \nu(\widetilde{L}), \partial(S^3 - \nu(\widetilde{L})))$, and $\chi(\widetilde{F}) = 2\chi(F)$.  Hence $x_{\widetilde{L}}([\widetilde{K}]) \leq 2x_L([K])$.


However, recall that $x_{L}([K]) + 1$ is exactly the breadth of the Alexander $A_1$ grading in $\widehat{\mathit{HFL}}(S^3, L)$, and therefore that the breadth of the $A_1$ grading in $\widetilde{\mathit{HFL}}(\mathcal D)$ is $x_{L}([K]) + 1 + (n_1 -1) = x_{L}([K]) + n_1$.  Similarly, the breadth of the $A_1$ grading in $\widehat{\mathit{HFL}}(S^3, \widetilde{L})$ is $x_{\widetilde{L}}([K]) +1$, and therefore the total breadth of the $A_1$ grading in $\widetilde{\mathit{HFL}}(\widetilde{\mathcal D})$ is $x_{\widetilde{L}}([K]) + 1 + 2n_1 - 1 = x_{\widetilde{L}}([\widetilde{K}]) + 2n_1$.  Moreover, we have seen that in the spectral sequence from $E^1 = \widetilde{\mathit{HFL}}(\widetilde{\mathcal D}) \otimes \mathbb Z_2((\theta))$ to $E^{\infty} \cong \widetilde{\mathit{HFL}}(\mathcal D) \otimes \mathbb Z_2((\theta))$, the relative $A_1$ grading of two elements on the $E^1$ page is twice the relative $A_1$ grading of their residues on the $E^{\infty}$ page.  Therefore the total breadth of the $A_1$ grading on the $E^1$ page of the spectral sequence is at least twice the breadth of the $A_1$ grading on the last page of the spectral sequence. We thus have the inequality
%
\begin{align*}
x_{\widetilde{L}}([\widetilde{K}]) + 2n_1 &\geq 2x_{L}([K]) + 2n_1 \\
										&\geq 2(x_L([K]) + n_1)\\
										\text{i.e. } x_{\widetilde{L}}([\widetilde{K}])&\geq 2x_L([K]).
\end{align*}
%
Consequently we see directly from the spectral sequence that
%
\begin{align*}
x_{\widetilde{L}}([\widetilde{K}]) = 2 x_L([K]).
\end{align*}
%
\noindent This implies that the breadth of the $A_1$ grading on the $E^1$ page is exactly twice the breadth of the $A_1$ grading on the $E^{\infty}$ page.  Therefore the breadth of the $A_1$ grading cannot decrease over the course of the spectral sequence.  A similar argument, sans the factors of two, shows that $x_{\widetilde L}([U]) = x_{L}([U])$ and that the breadths of the Alexander $A_2$ grading of the $E^1$ and $E^{\infty}$ pages of the spectral sequence are the same.  Therefore the breadth of the $A_2$ grading does not change either over the course of the spectral sequence.


\medskip

In particular, the top $A_1$ grading in $\widetilde{\mathit{HFL}}(\widetilde{\mathcal D})$ is sent to the top $A_1$ grading in $\widetilde{\mathit{HFL}}(\mathcal D)$.  However, by the symmetry of $\widehat{\mathit{HFL}}$ and the determination of the breadth of the $A_1$ grading by the Thurston norm, the top $A_1$ grading of $\widetilde{\mathit{HFL}}(\widetilde{\mathcal D})$ is the same as the top $A_1$ grading of $\widehat{\mathit{HFL}}(S^3, \widetilde{L})$, which is $A_1 = \frac{x_{\widetilde{L}}([\widetilde{K}]) +1}{2} = \frac{2x_L(K) +1}{2}$.  Similarly, the top $A_1$ grading of $\widetilde{\mathit{HFL}}(\mathcal D)$ is the top $A_1$ grading of $\widehat{\mathit{HFL}}(S^3, L)$, to wit, $A_1 = \frac{x_L([K]) + 1}{2}$.  Therefore the spectral sequence carries the $A_1$ Alexander grading $\frac{2x_L([K]) + 1}{2}$ on the $E^1$ page to the $A_1$ grading $\frac{x_L([K]) + 1}{2}$ on the $E^{\infty}$ page. Therefore in general the $A_1$ grading $\frac{2x_L([K]) + 1}{2} + 2a$ on the $E^1$ page is sent to the $A_1$ grading $\frac{x_L([K]) +1}{2} + a$ for any integer $a$.  Notice that $x_L([K])$ is an even number: suppose $F$ is a Thurston-seminorm minimizing surface for $K$ in $S^3-L$ of genus $g'$ with geometric intersection number $\#(F\cap U) =\Lambda$.  Since the algebraic intersection number $\lambda$ is odd, so is $\Lambda$.  Then $x_L([K]) = 1-2g'-\Lambda$ is even.  Therefore $\frac{x_L([K])}{2}$ is an integer, and we can take $a= \frac{x_L([K])}{2}$ to see that the $A_1$ grading $\frac{1}{2}$ on the $E^1$ page is sent to the $A_1$ grading $\frac{1}{2}$ on the $E^{\infty}$ page.

\medskip

A parallel but simpler argument for the $A_2$ gradings shows that the $A_2$ grading $b$ on the $E^1$ page is sent precisely to the $A_2$ grading $b$ on the $E^{\infty}$ page, using the fact that relative $A_2$ gradings of elements on the $E^1$ page that survive in the spectral sequence are preserved rather than doubled on the $E^{\infty}$ page.

\begin{remark} The observations that $x_{\widetilde{L}}([\widetilde{K}]) = 2 x_L([K])$ and $x_{\widetilde L}([U]) = x_{L}([U])$ are a special case of Gabai's theorem \cite[Corollary 6.13]{MR723813} that the Thurston norm is multiplicative for ordinary finite covers.  Indeed, by appealing to Gabai's theorem (or by constructing a $\mathbb Z_p$ analog of Seidel and Smith's localization spectral sequence) we could similarly fix the relationship between the absolute gradings of $\widetilde{\mathit{HFL}}(\widetilde{\mathcal D})$ and $\widetilde{\mathit{HFL}}(\mathcal D)$ for $p$-periodic knots.
\end{remark}

\begin{remark}

The reader may wonder at the asymmetry in the construction that causes the $A_1$ gradings $\frac{1}{2}$ and $\frac{1}{2}$ to correspond, but not the gradings $-\frac{1}{2}$ and $-\frac{1}{2}$.  This is the result of the convention that the two summands of the vector space $V_1$ have $A_1$ gradings $0$ and $-1$.

\medskip

This serves to illustrate the important role of the vector space $V_1$ in the existence of the spectral sequence; the breadth of the $A_1$ grading in $\widehat{\mathit{HFL}}(S^3, \widetilde{L})$ is one less than twice the breadth of the $A_1$ grading in $\widehat{\mathit{HFL}}(S^3, L)$, a trouble which is corrected for by increasing the breadth upstairs by $2n_1 - 1$ and the breadth downstairs by $n_1 -1$.  While one might hope to produce a spectral sequence in which $n_1 =1$, it seems impossible to produce a link Floer homology spectral sequence for doubly periodic knots which does not involve at least one copy of $V_1$ on the $E^1$ page.

\end{remark}

We summarize the discussion above in the following two lemmas.


\begin{lemma} \label{Alexander Gradings Link Lemma}

The spectral sequence from $E^1 = \widetilde{\mathit{HFL}}(\widetilde{\mathcal D}) \otimes \mathbb Z_2((\theta))$ to $E^{\infty} \cong \widetilde{\mathit{HFL}}(\mathcal D) \otimes \mathbb Z_2((\theta))$ splits along Alexander gradings. For $a \in \mathbb Z, b \in \mathbb Z+\frac{1}{2}$, the sequences sends the gradings $(A_1, A_2) = (\frac{1}{2} + 2a, b)$ on the $E^{1}$ page to the gradings $(A_1, A_2) = (\frac{1}{2} + a, b)$ on the $E^{\infty}$ page and kills all other gradings.  In particular there is a rank inequality
%
\begin{align*}
rk\left(\widetilde{\mathit{HFL}}\left(\widetilde{\mathcal D}, \left(\frac{1}{2} +2a, b\right)\right)\right) \geq rk\left(\widetilde{\mathit{HFL}}\left(\mathcal D, \left(\frac{1}{2}+a, b\right)\right)\right).
\end{align*}

\end{lemma}

By Lemma \ref{Gradings Shift Lemma}, computing the knot Floer homology complex using $\partial_U$ yields a downward shift in Alexander gradings by $\frac{\ell k(\widetilde{K}, U)}{2}$ on the $E^1$ page and $\frac{\ell k(K, U)}{2}$ on the $E^{\infty}$ page.  Since both of these numbers are $\frac{\lambda}{2}$, we obtain an overall downward shift of $\frac{\lambda}{2}$ between that link and knot Floer homology spectral sequences, leading to the following lemma.



\begin{lemma} \label{Alexander Gradings Knot Lemma}

The spectral sequence from $E^1 = \widetilde{\mathit{HFK}}(\widetilde{\mathcal D}) \otimes \mathbb Z_2((\theta))$ to $E^{\infty} \cong \widetilde{\mathit{HFK}}(\mathcal D) \otimes \mathbb Z_2((\theta))$ splits along the Alexander grading. For $a \in \mathbb Z$, the sequences sends the gradings $A_1 = \frac{1 -\lambda}{2} + 2a$ on the $E^{1}$ page to the gradings $A_1 = (\frac{1-\lambda}{2} + a)$ on the $E^{\infty}$ page and kills all other gradings.  In particular there is a rank inequality
%
%
\begin{align*}
rk\left(\widetilde{\mathit{HFK}}\left(\widetilde{\mathcal D}, 2a + \frac{1 - \lambda}{2}\right)\right) \geq rk\left(\widetilde{\mathit{HFK}}\left(\mathcal D, a + \frac{1 - \lambda}{2}\right)\right).
\end{align*}

\end{lemma}





Having fixed the Alexander gradings in the spectral sequence, we may provide a proof of Corollary \ref{Genus Inequality Corollary} (Edmonds' Condition) from \ref{Knot Floer Homology Spectral Sequence}.



\begin{proof} [Proof of Corollary \ref{Genus Inequality Corollary}]

Once again, let $V_1$ denote a two-dimensional vector space over $\mathbb F_2$ whose two sets of gradings $(M, (A_1, A_2))$ are $(0,(0,0))$ and $(-1, (-1, 0))$, and likewise let $W$ be a two-dimensional vector space over $\mathbb F_2$ whose two sets of gradings $(M, (A_1,A_2))$ are $(0,(0,0))$ and $(-1,(0, 0))$.  By Theorem \ref{Knot Floer Homology Spectral Sequence}, there is a spectral sequence whose $E^1$ page is $\widehat{\mathit{HFK}}(S^3, \widetilde{K}) \otimes V_1^{\otimes n_1-1} \otimes W \otimes \mathbb Z_2((\theta))$ to a theory whose $E^{\infty}$ page is $\mathbb Z((\theta))$-isomorphic to $\widehat{\mathit{HFK}}(S^3, K) \otimes V_1^{\otimes 2n_1-1} \otimes W \otimes \mathbb Z((\theta))$.  Moreover, this spectral sequence splits along the $A_1$ grading, and by Lemma \ref{Alexander Gradings Knot Lemma} the subgroup of the $E^1$ page in $A_1$ grading $\frac{1-\lambda}{2} + 2a$ is carried to the subgroup of the $E^{\infty}$ page in grading $\frac{1-\lambda}{2} +a$.  Moreover, the top $A_1$ grading on the $E^1$ page is $g(\widetilde{K})$ and the top $A_1$ grading on the $E^{\infty}$ page is $g(K)$.  Since there must be something on the $E^1$ page in the $A_1$ grading which converges to the $A_1$ grading $g(K)$ on the $E^{\infty}$ page, we have the following inequality.
%
\begin{align*}
g(\widetilde{K}) - \frac{1-\lambda}{2} &\geq 2\left(g(K) - \frac{1 - \lambda}{2}\right) \\
\text{i.e. } g(\widetilde{K}) &\geq 2g(K) + \frac{\lambda - 1}{2} \qedhere
\end{align*}

\end{proof}

Finally, let us prove Corollary \ref{Fiberedness Corollary}.

\begin{proof}[Proof of Corollary \ref{Fiberedness Corollary}]

Suppose Edmonds' condition is sharp, that is, that $g(\widetilde{K}) = 2g(K) + \frac{\lambda - 1}{2}$.  Recall that the spectral sequence from $E^1 \cong \widetilde{\mathit{HFK}}(\widetilde{\mathcal D}) \otimes \mathbb Z_2((\theta))$ to $E^{\infty} \cong \widetilde{\mathit{HFK}}(\mathcal D) \otimes \mathbb Z_2((\theta))$ in general sends the Alexander grading $2a + \frac{1-\lambda}{2}$ to the Alexander grading $a+\frac{1-\lambda}{2}$.  Therefore it sends the Alexander grading $g(\widetilde{K}) = 2g(K) + \frac{\lambda -1}{2} = 2g(K) + \frac{\lambda-1}{2}) + \frac{1-\lambda}{2}$ to the Alexander grading $g(K)$.  That is, sharpness of Edmonds' condition exactly says that the top Alexander grading on the $E^1$ page is not killed in the spectral sequence.

\medskip

Suppose now that $\widetilde{K}$ is fibered.  Then the top Alexander grading of $\widetilde{\mathit{HFK}}(\widetilde{\mathcal D}) \otimes \mathbb Z_2((\theta)) = (\widehat{\mathit{HFK}}(S^3, L) \otimes V_1^{\otimes (2n_1-1)}\otimes W) \otimes \mathbb Z_2((\theta))$ has rank two as a $\mathbb Z_2((\theta))$ module.  (The knot Floer homology of fibered knots is monic in the top Alexander grading by the forward direction of  Lemma \ref{Fiberedness Condition Lemma} , and the factor of $W$ doubles the number of entries in each Alexander grading.)  Since this Alexander grading is not killed in the spectral sequence, the top Alexander grading of $\widetilde{\mathit{HFK}}(\mathcal D) \otimes \mathbb Z_2((\theta))=\widehat{\mathit{HFK}}(S^3, L) \otimes V_1^{\otimes (n_1-1)} \otimes W) \otimes \mathbb Z_2((\theta))$ also has rank two as a $\mathbb Z_2((\theta))$-module.  Therefore $K$ is also fibered.\end{proof}

\begin{remark}

The converse of Corollary \ref{Fiberedness Corollary}, that when Edmonds' condition is sharp, the quotient knot $K$ being fibered implies $\widetilde{K}$ is fibered, is false.  Consider the following counterexample: the knot $\widetilde{K}=10_{144}$ is doubly periodic with quotient knot $K = 3_1$, the trefoil.  The linking number $\lambda = \ell k(\widetilde{K}, U) = 1$.  Since $g(10_{144}) = 2$ and $g(3_1)=1$, we see that $g(\widetilde{K}) = 2 = 2g(K) + \frac{\lambda-1}{2}$.  Therefore Edmonds' condition is sharp.  However, the trefoil is fibered, whereas $10_{144}$ is not.



\end{remark}




\section{Spectral Sequences for Lagrangian Floer Cohomology} \label{Floer Cohomology Section}

Floer cohomology is an invariant for Lagrangian submanifolds in a symplectic manifold introduced by Floer \cite{MR965228, MR933228, MR948771}. Many versions of the theory exist; in this section we briefly introduce Seidel and Smith's setting for Floer cohomology before stating the hypotheses and results of their main theorem for equivariant Floer cohomology. Let $M$ be a manifold equipped with an exact symplectic form $\omega = d\theta$ and a compatible almost complex structure $J$. Let $L_0$ and $L_1$ be two exact Lagrangian submanifolds of $M$. For our purposes we can restrict to the case that $L_0$ and $L_1$ intersect transversely. 

\begin{definition}
The Floer chain complex $\mathit{CF}(L_0,L_1)$ is a $\mathbb Z_2$-vector space with generators the finite set of points $L_0 \cap L_1$.
\end{definition}

The differential $d$ on $\mathit{CF}(L_0,L_1)$ counts holomorphic disks whose boundary lies in $L_0 \cup L_1$ which run from $x_-$ to $x_+$. More precisely, we choose ${\bf J} = J_t$ a time-dependent perturbation of $J$ and consider Floer trajectories $u$ of the following form.

\begin{align*}
u: \mathbb R \times [0,1] \rightarrow M \\
u(s,0) \in L_0, u(s,1) \in L_1 \\
\partial_s u + J_t\partial_t(u) =0 \\
\lim_{s\rightarrow \pm \infty} = x_{\pm} \\
\end{align*}

The moduli space of such maps carries a natural action by $\mathbb R$ corresponding to translation on the coordinate $s$; we let the quotient by this action be $\mathcal M(x_{-}, x_{+})$, and the compactification of this space be $\widehat{\mathcal M}(x_{-}, x_{+})$.

\medskip

We impose one further technical condition on $M$ to ensure both that there are finitely many equivalence classes of holomorphic curves between any two intersection points $x_+,x_{-} \in L_0 \cap L_1$ and that the image of any holomorphic curve $u:\mathbb R \times [0,1] \rightarrow M$ is contained in some compact set in $M$. We say $\phi: M \rightarrow \mathbb R$ is \textit{exhausting} if it is smooth, proper, and bounded below. We consider the one-form $d^{\mathbb C}(\phi) = d\phi \circ J$ and the two-form $\omega_{\phi} = -dd^{\mathbb C}(\phi)$. We say that $\phi$ is \textit{$J$-convex} or \textit{plurisubharmonic} if $\omega$ is compatible with the complex structure on $M$, that is, if $\omega(Jv,Jw) = \omega(v,w)$ and  $\omega(v,Jv) > 0$ for all $v,w \in TM$. This ensures that $\omega_{\phi}$ is a symplectic form on $M$.(The term plurisubharmonic indicates that the restriction of $\phi$ to any holomorphic curve in $M$ is subharmonic, hence satisfies the maximum modulus principle.) A noncompact symplectic manifold $M$ with this structure is called \textit{convex at infinity}. 

\medskip


We use an index condition to determine which strips $u$ count for the differential. Given any $u 
\in \mathcal M (x_-,x_+)$, we can associate to $u$ a Fredholm operator $D_{{\bf J}}u: \mathcal W^1_u \to \mathcal W^0_u$ from $\mathcal W^1_u = \{X \in W^{1,p}(u^*TM) : X(\cdot, 0) \in u^*TL_0, X(\cdot, 1) \in u^*TL_1\}$ to $W^0_u$. (Here $p>2$ is a fixed real number.) This operator describes the linearization of Floer's equation, $\partial_s u + J_t(u)\partial_t(u) =0$, near $u$. We say that ${\bf J}$ is \textit{regular} if $D_{{\bf J}} u$ is surjective for all finite energy holomorphic strips $u$.

\begin{definition}
The index of the operator $D_{{\bf J}} u$ is the \textit{Maslov index} of $u$.
\end{definition}

\begin{lemma}
If $M$ is an exact symplectic manifold with a compatible almost convex structure $J$ which is convex at infinity and $L_0,L_1$ are exact Lagrangian submanifolds also convex at infinity, then a generic choice of ${\bf J}$ perturbing $J$ is regular.
\end{lemma}



Floer's original proof of this result \cite[Proposition 2.1]{MR965228} and Oh's revision \cite[Proposition 3.2]{MR1223659} were for compact manifolds, but, as observed in \cite[Section 9.2]{MR2045629} and indeed by Sikorav \cite{Sikorav} in his review of Floer's paper \cite{MR965228}, the proof carries through identically for noncompact manifolds which are convex at infinity. Choose such a generic regular ${\bf J}$. We let $\mathcal M_1(x_{-}, x_{+})$ be the set of classes of trajectories $u$ in $\mathcal M(x_{-},x_{+})$ such that the Fredholm index of $D_{{\bf J}}u$ is $1$. 

\begin{lemma}\cite[Lemma 3.2]{MR965228}
If ${\bf J}$ is regular, then $\mathcal M_1(x_{-},x_{+})$ is a smooth, compact 0-manifold such that $\#\widehat{ \mathcal M}_1(x_{-}, x_{+})$ is finite. Moreover, for any $x_-, x_+ \in \mathit{CF}(L_0,L_1)$, the sum
%
\[
\sum_{x \in \mathit{CF}(L_0,L_1)}\#\widehat{ \mathcal M}_1(x_{-}, x)\#\widehat{ \mathcal M}_1(x_, x_{+})
\]
%
\noindent is zero modulo two. 

\end{lemma}


Therefore we make the following definition.


\begin{definition} \label{FloerCohomologyDefn}\cite[Defn 3.2]{MR965228}
The Floer cohomology $\mathit{HF}(L_0,L_1)$ is the homology of $\mathit{CF}(L_0,L_1)$ with respect to the differential 
%
\begin{align} 
\delta(x_{-}) = \sum_{x_+ \in \mathit{CF}(L_0,L_1)} \# \widehat{\mathcal M_1}(x_{-},x_{+})x_+ \label{differential}
\end{align}
%
\noindent with respect to a regular family of almost complex structures ${\bf J}$ perturbing $J$. 

\end{definition}


Now suppose that $M$ carries a symplectic involution $\tau$ preserving $(M, L_0, L_1)$ and the forms $\omega$ and $\theta$. Let the submanifold of $M$ fixed by $\tau$ be $M^{\text{inv}}$, and similarly for $L_i^{\text{inv}}$ for $i=0,1$. We can define the Borel (or equivariant) cohomology of $(M, L_0, L_1)$ with respect to this involution. Seidel and Smith give a geometric description of the cochain complexes used to produce equivariant Floer cohomology; we'll content ourselves with an algebraic description, referring the reader to their paper \cite[Section 3]{MR2739000} for further geometric detail. Notice that the usual Floer chain complex $\mathit{CF}(L_0, L_1)$ carries an induced involution $\tau^{\#}$ which takes an intersection point $x \in L_0 \cap L_1$ to the intersection point $\tau(x) \in L_0 \cap L_1$. This map $\tau^{\#}$ is not a chain map with respect to a generic family of complex structures on $M$.  However, suppose that we are in the nice case that we can find a suitable family of complex structures $J$ on $M$ such that $\tau^{\#}$ commutes with the differential on $\mathit{CF}(L_0, L_1)$. (Part of Seidel and Smith's use of their technical conditions on the bundle $\Upsilon(M^{\text{inv}})$ is to establish that such a $J$ exists \cite[Lemma 19]{MR2739000}.) Then $\mathit{CF}(L_0, L_1)$ is a chain complex over $\mathbb F_2[\mathbb Z_2] = \mathbb F_2[\tau^{\#}]/\langle (\tau^{\#})^2 = 1\rangle$. Indeed, $(1 + \tau^{\#})^2 = 0$, so there is a chain complex
%
%
$$
\xymatrix{
0 \rightarrow \mathit{CF}(L_0, L_1) \ar[r]^-{1 + \tau^{\#}} & \mathit{CF}(L_0, L_1) \ar[r]^-{1 + \tau^{\#}} & \mathit{CF}(L_0, L_1)\cdots
}
$$
%
%
\begin{definition}

If $\mathit{CF}(L_0, L_1)$ is the Floer chain complex and $\tau^{\#}$ is a chain map with respect to the complex structure on $M$, $\mathit{HF}_{\text{borel}}(L_0, L_1)$ is the homology of the complex $\mathit{CF}(L_0,L_1) \otimes \mathbb Z[[\theta]]$ with respect to the differential $\delta + (1 + \tau^{\#})q$. 

\end{definition}




Therefore the double complex
%
%
\label{doublecomplex}
$$
\xymatrix{
0 \rightarrow \mathit{CF}(L_0,L_1) \ar@(ur,ul)[]_-{\delta} \ar[r]^-{1+\tau^{\#}} & \mathit{CF}(L_0, L_1) \ar[r]^-{1+\tau^{\#}} \ar@(ur,ul)[]_-{\delta} & \mathit{CF}(L_0, L_1)\cdots \ar@(ur,ul)[]_-{\delta}
}
$$
%
%
\noindent induces a spectral sequence whose first page is $\mathit{HF}(L_0,L_1) \otimes \mathbb Z_2[[\theta]]$ and which converges to $\mathit{HF}_{\text{borel}}(L_0,L_1)$.
 


\medskip

We also have the following more algebraic definition of equivariant Floer cohomology.  The following lemma is well-known; a proof may be found in \cite[Section 2]{Hendricks}.

\begin{lemma} \label{homspace}

\noindent The equivariant Floer cohomology $\mathit{HF}_{\text{borel}}(L_0,L_1)$ is isomorphic to
%
\[
Ext_{\mathbb F_2[\mathbb Z_2]}(\mathit{CF}(L_1,L_0), \mathbb F_2).
\]
%
\end{lemma}


Here we regard $\mathbb F_2$ as the trivial module over $\mathbb F_2[\mathbb Z_2]$.







\medskip

Seidel and Smith's result concerns the existence of a localization map from $\mathit{HF}_{\text{borel}} (L_0, L_1)$ to $\mathit{HF}(L_0^{\text{inv}}, L_1^{\text{inv}})$, where the second space is the Floer cohomology of the two Lagrangians $L_0^{\text{inv}}$ and $L_1^{\text{inv}}$ in $M^{\text{inv}}$. The main goal is to produce a family of $\tau$-invariant complex structures on $M$ such that, for $u: \mathbb R \times [0,1] \rightarrow M^{\text{inv}}$, the Maslov index of $u$ in $M^{\text{inv}}$ differs from the Maslov index of $u$ in $M$ by a constant.

\medskip

Consider the normal bundle $N(M^{\text{inv}})$ to $M^{\text{inv}}$ in $M$ and its Lagrangian subbundles $N(L_i^{\text{inv}})$ the normal bundles to each $L_i^{\text{inv}}$ in $L_i$. The construction requires one additional degree of freedom, achieved by pulling back the bundle $N(M^{\text{inv}})$ along the projection map $M^{\text{inv}} \times [0,1] \rightarrow M^{\text{inv}}$. Call this pullback $\Upsilon(M^{\text{inv}})$. This bundle is constant with respect to the interval $[0,1]$. Its restriction to each $M^{\text{inv}} \times \{t\}$ is a copy of $N(M^{\text{inv}})$ which will occasionally, by a slight abuse of notation, be called $N(M^{\text{inv}}) \times \{t\}$; similarly, for $i=0,1$ the copy of $N(L_i^{\text{inv}})$ above $L_i^{\text{inv}} \times \{t\}$ will be referred to as $N(L_i^{\text{inv}}) \times \{t\}$.

\medskip

We make a note here of the correspondence between our notation and Seidel and Smith's original usage. Our bundle $\Upsilon(M^{\text{inv}})$ is their $TM^{\text{anti}}$; while our $N(L_0^{\text{inv}}) \times \{0\}$ is their $TL_0^{\text{inv}}$ and our $N(L_1^{\text{inv}}) \times \{1\}$ is their $TL_1^{\text{anti}}$. (The name $TL_1^{\text{anti}}$ is also used for the bundle that we denote $N(L_1^{\text{inv}}) \times \{0\}$, using the obvious isomorphism between the bundles.)

\medskip

\begin{definition} \label{stablenormaltriv} \cite[Defn 18]{MR2739000} A stable normal trivialization of the vector bundle $\Upsilon(M^{\text{inv}})$ over $M^{\text{inv}} \times [0,1]$ consists of the following data. 


\begin{itemize}
\item A stable trivialization of unitary vector bundles $\phi: \Upsilon(M^{\text{inv}}) \oplus \mathbb C^{K} \rightarrow \mathbb C^{k_{\text{anti}} + K}$ for some $K$.

\item A Lagrangian subbundle $\Lambda_0 \subset (\Upsilon(M^{\text{inv}}))|_{[0,1] \times L^{\text{inv}}_0}$ such that $\Lambda_0|_{\{0\} \times L^{\text{inv}}_0} = (N(L_0^{\text{inv}})\times \{0\})\oplus \mathbb R^K$ and $\phi(\Lambda_0|_{\{1\} \times L_0^{\text{inv}}}) = \mathbb R^{k_{\text{anti}} + K}$.

\item A Lagrangian subbundle $\Lambda_1 \subset (\Upsilon(M^{\text{inv}}))|_{[0,1] \times L^{\text{inv}}_1}$ such that $\Lambda_1|_{\{0\} \times L^{\text{inv}}_1} = (N(L_1^{\text{inv}})\times\{0\})\oplus \mathbb R^K$ and $\phi(\Lambda_1|_{\{1\} \times L_1^{\text{inv}}}) = i\mathbb R^{k_{\text{anti}} + K}$.

\end{itemize}


\end{definition}

The crucial theorem of \cite{MR2739000}, proved through extensive geometric analysis and comparison with the Morse theoretic case, is as follows.


\begin{theorem} \label{Localization} \cite[Thm 20]{MR2739000} If $\Upsilon(M^{\text{inv}})$ carries a stable normal trivialization, then $\mathit{HF}_{\text{borel}}(L_0,L_1)$ is well-defined and there are localization maps
%
\[
\Delta^{(m)}: \mathit{HF}_{\text{borel}} \rightarrow \mathit{HF}(L_0^{\text{inv}}, L_1^{\text{inv}})[[\theta]]
\]
%
\noindent defined for $m>>0$ and satisfying $\Delta^{(m+1)} = \theta\Delta^{(m)}$. Moreover, after tensoring over $\mathbb Z_2[[\theta]]$ with $\mathbb Z_2((\theta))$ these maps are isomorphisms. 

\end{theorem}


This implies Theorem \ref{SeidelSmith}. 


\medskip

We can in fact dispense with the symplectic structure and work on the level of the complex normal bundle $\Upsilon(M^{\text{inv}})$ with its totally real subbundles $NL_0^{\text{inv}} \times \{1\}$ and $J(NL_1^{\text{inv}} \times \{1\})$.  The following lemma is mentioned in \cite[Section 3d]{MR2739000}; a detailed proof is laid out in \cite[Proposition 7.1]{Hendricks}.

\begin{lemma} \label{Nullhomotopy Lemma}

The existence of a stable normal trivialization of $(M, L_0, L_1)$ is implied by the existence of a nullhomotopy of the map
%
\begin{align*}
(M, L_0, L_1) \rightarrow (BU, BO)
\end{align*}
%
\noindent which classifies the complex normal bundle $\Upsilon(M^{\text{inv}}) = NM^{\text{inv}} \times [0,1]$ and its totally real subbundles $NL_0^{\text{inv}} \times \{0\}$ over $L_0^{\text{inv}} \times \{0\}$ and $J(NL_1^{\text{inv}}) \times \{1\}$ over $L_1^{\text{inv}} \times \{1\}$.

\end{lemma}


Let $\widetilde{\mathcal D} = (S^2, \widetilde{\boldsymbol \alpha}, \widetilde{\boldsymbol \beta}, \widetilde{\bf w}, \widetilde{\bf z})$ be a multipointed Heegaard diagram for $\widetilde{K}$ defined using the method of Section \ref{Periodic Knots Section}, and $\mathcal D = (S^2, {\boldsymbol \alpha}, {\boldsymbol \beta}, {\bf w}, {\bf z})$ be its quotient under the involution $\tau$.  Given $x$ a point on $D$, let $x^1,x^2$ be its two lifts to $\widetilde{\mathcal D}$ in some order.  There is a natural map
%
\begin{align*}
\iota:\text{Sym}^{n_1}(S^2) &\rightarrow \text{Sym}^{2n_1}(S^2) \\
(x_1\cdots x_{n_1}) &\mapsto (x_1^1x_1^2\cdots x_{n_1}^1x_{n_1}^2).
\end{align*}

This map is a holomorphic embedding; for a proof, see \cite[Appendix 1]{Hendricks}.  Moreover, consider the induced involution on $\text{Sym}^{2n_1}(S^2)$, which through a slight abuse of notation we will also call $\tau$.  The fixed set of $\tau$ is exactly our embedded copy of $\text{Sym}^{n_1}(S^2)$; moreover, $\tau$ preserves the two tori $\mathbb T_{\widetilde{\boldsymbol \alpha}}$ and $\mathbb T_{\widetilde{\boldsymbol \beta}}$, with fixed sets $T_{\widetilde{\boldsymbol \alpha}}^{\text{inv}} = \mathbb T_{\boldsymbol \alpha}$ and $T_{\widetilde{\boldsymbol \beta}}^{\text{inv}} = \mathbb T_{\boldsymbol \beta}$.

\medskip



Perutz has shown that for an arbitrary Heegaard diagram $D = (S, \boldsymbol \alpha, \boldsymbol \beta, {\bf w,z})$, there is a symplectic form $\omega$ on $\text{Sym}^{g+n-1}(S)$  which is compatible with the complex structure induced by a complex structure on $S$, and with respect to which the submanifolds $\mathbb T_{\alpha}$ and $\mathbb T_{\beta}$ are in fact Lagrangian and the various Heegaard Floer homology theories are their Lagrangian Floer cohomologies \cite[Thm 1.2]{MR2509747}. In particular, the knot Floer homology is the Floer cohomology of these two tori in the ambient space $\text{Sym}^{g+n-1}(S \backslash \{{\bf z,w}\})$, where the removal of the basepoints accounts for the restriction that holomorphic curves not be permitted to intersect the submanifolds $V_{w_i}$ and $V_{z_j}$ of the symmetric product.


\begin{proposition}
There is a symplectic structure on $\text{Sym}^{g-n-1}(S^2 \backslash \{{\bf w,z}\})$ with respect to which the submanifolds $\mathbb T_{\boldsymbol \alpha}$ and $\mathbb T_{\boldsymbol \beta}$ are Lagrangian and

\[
\widetilde{\mathit{HFL}}(D) \cong \widehat{\mathit{HFK}}((S^3,K))\otimes V_1^{\otimes (n_1-1)}\otimes\cdots\otimes V_{\ell}^{\otimes (n_{\ell}-1)} \cong \mathit{HF}(\mathbb T_{\boldsymbol \beta}, \mathbb T_{\boldsymbol \alpha}).
\]

\end{proposition}

This is essentially Theorem 1.2 of \cite{MR2509747}; the proof is identical safe for trivial notation adjustments for Heegaard diagrams with multiple basepoints. From now on we will work with this method of computing knot Floer homology; in Section \ref{Geometry Section} we will show that the symplectic form produced by Perutz's construction meets the requirements of Seidel and Smith's theorem.

\medskip

In order to apply Theorem \ref{Localization} to the case of doubly periodic knots, we will work with three different subspaces of $\text{Sym}^{2n_1}(S^2)$ and their fixed sets under the involution $\tau$, as follows.
%
\begin{align*}
M_0 &= \text{Sym}^{2n_1}(S^2 \backslash \{\widetilde{{\bf w}}\}) && 
M_0^{\text{inv}} = \text{Sym}^{n_1}(S^2 \backslash \{{\bf w}\}) \\
M_1 &= \text{Sym}^{2n_1}(S^2 \backslash \{{\bf \widetilde{w}}, {\bf \widetilde{z}}-z_0\}) &&
M_1^{\text{inv}} = \text{Sym}^{n_1}(S^2 \backslash \{{\bf w}, {\bf z}-z_0 \}) \\
M_2 &= \text{Sym}^{2n_1}(S^2 \backslash \{\widetilde{\bf w}, \widetilde{\bf z}\}) &&
M_2^{\text{inv}}= \text{Sym}^{n_1}(S^2 \backslash \{{\bf w,z}\})
\end{align*}



\noindent In all cases the Lagrangians and their invariant sets under the involution will be as follows.
%
\begin{align*}
L_0 = \mathbb T_{\widetilde{\boldsymbol \beta}} && L_0^{\text {inv}} = \mathbb T_{\boldsymbol \beta} \\
L_1 = \mathbb T_{\widetilde{\boldsymbol \alpha}} && L_1^{\text {inv}} = \mathbb T_{\boldsymbol \alpha}
\end{align*}

The following is immediate from the definitions.

\begin{lemma}

With respect to our choice of symplectic manifolds $M_i$ for $i=1,2,3$ and Lagrangians $L_0$ and $L_1$, we have the following Floer cohomology groups.

\noindent In $M_1$,
%
\begin{align*}
\mathit{HF}(L_0,L_1) &= \mathit{HF}(\mathbb T_{\widetilde{\boldsymbol \beta}}, \mathbb T_{\widetilde{\boldsymbol \alpha}}) = \widehat{\mathit{HFL}}(S^3, \widetilde{K} \cup U) \otimes V^{\otimes (2n_1-1)}.\\
\mathit{HF}(L_0^{\text{inv}}, L_1^{\text{inv}})&=\mathit{HF}(\mathbb T_{\boldsymbol \beta}, \mathbb T_{\boldsymbol \alpha}) = \widehat{\mathit{HFL}}(S^3, K \cup U) \otimes V^{\otimes (n_1-1)}.
\end{align*}

\noindent In $M_2$,
%
\begin{align*}
\mathit{HF}(L_0,L_1)&=\mathit{HF}(\mathbb T_{\widetilde{\boldsymbol \beta}}, \mathbb T_{\widetilde{\boldsymbol \alpha}}) = \widehat{\mathit{HFK}}(S^3, \widetilde{K})\otimes V^{\otimes (2n_1-1)}\otimes W.\\
\mathit{HF}(L_0^{\text{inv}},L_1^{\text{inv}})&=\mathit{HF}(\mathbb T_{\boldsymbol \beta}, \mathbb T_{\boldsymbol \alpha}) = \widehat{\mathit{HFK}}(S^3,K) \otimes V^{\otimes (n_1-1)}\otimes W.
\end{align*}


\end{lemma}


We will prove in Section \ref{Stable Normal Triv Section} that the space $(M_0^{\text{inv}}, L_0^{\text{inv}}, L_1^{\text{inv}})$ carries a stable normal trivialization; this then naturally restricts to a stable normal trivialization of $(M_i^{\text{inv}}, L_0^{\text{inv}}, L_1^{\text{inv}})$ for $i=1,2$. (These manifolds will carry slightly different symplectic structures, but by Lemma \ref{Nullhomotopy Lemma} we need only be concerned with their complex structures, which will be determined by the inclusion $M_i^{\text{inv}} \subset M_0^{\text{inv}}$.) In some moral sense, this is the correct level of generality: the most important feature of our punctured Heegaard diagram is that no periodic domain has nonzero index, and ${\bf w}$ (or ${\bf z}$) is the smallest set of points at which we may puncture $S^2$ and produce a diagram for which this property holds.  Moreover, $M_0^{\text{inv}}$ conveniently deformation retracts onto each of $\mathbb T_{\boldsymbol \alpha}$ and $\mathbb T_{\boldsymbol \beta}$, making certain cohomology computations in Sections \ref{Geometry Section} and \ref{Stable Normal Triv Section} cleaner than they might otherwise be.

\medskip

An important note is that the localization map of Theorem \ref{Localization} is entirely constructed by counting pseudoholomorphic disks of varying index and by multiplication and division by powers of $\theta$.  In particular,  on $M_1^{\text{inv}}$, the localization isomorphism preserves the Alexander multigrading $(A_1,A_2)$ on th $E^{\infty}$ page, because no flowlines can pass over the missing basepoint divisors $V_{z_i}$ and $V_{w_i}$ for $0\leq i\leq n_1$, and on $M_2^{\text{inv}}$, the localization isomorphism preserves the Alexander grading $A_1$ since flowlines can pass over $V_{z_0}$ and $V_{w_0}$ but no other basepoint divisors.  In general, however, we should expect that these isomorphisms will not preserve the data of the Maslov grading.

\section{The Geometry of the Symmetric Product} \label{Geometry Section}

It remains to be verified that $M_0^{\text{inv}}$, $M_1^{\text{inv}}$, and $M_2^{\text{inv}}$ satisfy the basic symplectic structural requirements of Seidel and Smith's theory.  To wit, we must see that these manifolds can be equipped with triples $(j_i, \omega_i, \theta_i)$ for $i=1,2,3$ such that $\omega_i = d \theta_i$, $\theta_i|_{L_k^{\text{inv}}}$ is exact for $k=0,1$, and moreover the complex structure $j_i$ is convex at infinity.

\medskip

The proof closely follows the methods of \cite[Section 4]{Hendricks}.  Give $S^2 \backslash \{w_0\}$ a complex structure $j$ by identifying it with the complex plane (such that $z_0$ lies at zero).  This induces a complex structure $\text{Sym}^{n_1}(j)$ on $\text{Sym}^{n_1}(S^2 \backslash \{w_0\})$ which restricts to complex structures $j_i = \text{Sym}^{n_1}(j)|_{M_i^{\text{inv}}}.$ For $i = 0,1,2$, let ${\bf r_i}$ be the finite set of points on $S^2$ such that $M_i = \text{Sym}^{2n_1}(S^2 \backslash \{{\bf r_i}\})$.  We will show that $M_i$ is convex at infinity and can be equipped with a symplectic form which is $\tau$-invariant and with respect to which $\mathbb T_{\boldsymbol \alpha}$ and $\mathbb T_{\boldsymbol \beta}$ are exact Lagrangians.

\medskip

Let $f_i$ be an exhausting function on $S^2 \backslash \{{\bf r_i}\}$ as follows. Let $x = u+iv$ be a complex number.
%
\begin{align*}
f_i: S^2 \backslash \{{\bf r_i}\} \rightarrow \mathbb R \\
x \mapsto C_i |x|^2 + \sum_{j=1}^{k_i} \frac{1}{|x-r_j|^2}
\end{align*}

Here $C_i>0$ is a constant determined as follows.  The exact symplectic form associated to $f_i$ is
%
\begin{align*}
\omega_{f_i} = -dd^{\mathbb C_i}(f_i) = \left(4C_i + \sum_{j=1}^{k_i} \frac{4}{|x-r_i|^4}\right) du\wedge dv.
\end{align*}

The $\alpha$ and $\beta$ curves on $S^2$ are certainly Lagrangian with respect to the symplectic form $\omega_{f_i}$, but we require them to moreover be exact; that is, we wish to ensure that $\int_{\alpha_j^k} -d^{\mathbb C}(f_i) = 0$ for all $\alpha_j^k$ on $S^2 \backslash \{{\bf r_i}\}$. We choose $C_i$ large enough that the integral $\int_{\alpha_j} -d^{\mathbb C}f_i > 0$ for $1\leq j\leq n_1$, and similarly for each $\beta_j$.  Then we may isotope the $\alpha$ and $\beta$ curves inward toward the punctures ${\bf w}$ (around which the curves are oriented counterclockwise) until we have a new, isotopic set of curves for which $\int_{\alpha_j} -d^{\mathbb C}f_i = 0 = \int_{\alpha_j} -d^{\mathbb C}f_i$.  Hence we have arranged that the $\alpha$ and $\beta$ curves are exact Lagrangians with respect to $\omega_{f_i}$.

\medskip

On the product space $(S^2 \backslash \{{\bf r_i}\})^{2n_1}$ there is a corresponding exhausting function defined as follows.  Let $p_k$ be the projection of the product to its $k$th factor, and let $\widetilde{f_i} = f_i \circ p_1 + \cdots + f_i \circ p_{2n_1}$.  Notice that $\text{Sym}^{2n_1}(j_i)$-convexity, properness, and boundedness of $\widetilde{f_i}$ follow from the corresponding properties of $f_i$. The symplectic form we obtain from $\widetilde{f}_i$ is $\omega_{\widetilde{f}_i} = \omega_{f_i} \otimes 1 \cdots \otimes 1 + 1 \otimes \omega_{f_i} \otimes 1 \otimes\cdots \otimes 1 +\cdots+1\otimes\cdots\otimes \omega_{f_i}$.  Therefore any of the $(2n_1)!$ lifts of $\mathbb T_{\widetilde{\boldsymbol \alpha}}$ and $\mathbb T_{\widetilde{\boldsymbol \beta}}$ is an exact Lagrangian submanifold with respect to $\omega_{\widetilde{f_i}}$.

\medskip

Finally, consider the symmetric product $\text{Sym}^{2n_1}(S^2 \backslash \{{\bf r_i}\})$.  There is a (possibly singular) continuous function
%
%
\begin{align*}
\psi_i: \text{Sym}^{2n_1}(S^2 \backslash \{{\bf r_i}\}) &\rightarrow \mathbb R \\
(x_1\cdots x_{2n_1}) &\mapsto \sum_{\sigma \in S_{2n_1}} \widetilde{f_i}(x_{\sigma(1)},\cdots,x_{\sigma(2n_1)})
\end{align*}
%
%
\noindent which is smooth outside a neighborhood of the fat diagonal $\{(x_1\cdots x_{2n_1}) \in \text{Sym}^{2n_1}(S^2 \backslash \{{\bf r_i}\}): x_k = x_j \text{ for some } k\neq j\}$. Perutz observes \cite{MR2509747} that this function is strictly plurisubharmonic in the sense of non-smooth functions, that is, that the two-current $-dd^{\mathbb C}\psi_i$ is strictly positive. This gives us a continuous exhausting function on $\text{Sym}^{2n-2}(\Sigma(S^2) \backslash \{{\bf w,z}\})$.  We may apply the following lemma of Richberg \cite{MR0222334}, quoted in \cite[Lemma 3.10]{SteinBook}.

\begin{lemma}
Let $\psi$ be a continuous $J$-convex function on an integrable complex manifold $V$. Then for every positive function $h:V \rightarrow \mathbb R_+$, there exists a smooth $J$-convex function $\psi'$ such that $|\psi(x) - \psi'(x)| < h(x)$. If $\phi$ is already smooth on a neighborhood of a compact subset $A$, then we can achieve $\phi = \phi'$ on $A$.
\end{lemma}

In particular, we may take $h:\text{Sym}^{2n-2}(\Sigma(S^2) \backslash \{{\bf w,z}\})\rightarrow \mathbb R_+$ to be a constant function $h(x) = \epsilon$, and apply the lemma to our map $\psi_i$ and $h$.  Then we may produce $\psi_i':\text{Sym}^{2n_1}(S^2 \backslash \{{\bf r_i}\}) \rightarrow \mathbb R$ such that $|\psi_i'(x) - \psi_i(x)|<\epsilon$ and $\psi_i'$ is smooth and $\text{Sym}^{2n_1}(j_i)$-convex.  Moreover, since $\psi_i$ is bounded below and proper, $\psi_i'$ is also, since the two real valued functions differ by at most $\epsilon$.  Therefore $\psi_i'$ is an exhausting function on $M_i$, and $M_i$ is convex at infinity.

\medskip

We have yet to produce an appropriate symplectic form on $M_i$, which we will do using work of Perutz \cite{MR2509747}. We begin with a definition.  


\begin{definition}\cite[Defn 7.3]{MR2509747} Let $X$ be a complex manifold with complex structure $J$. A \emph{K\"ahler cocycle} on X is a collection $(U_k, \phi_k)_{k \in K}$, where $(U_k)_{k \in K}$ is an open cover of $X$ and $\phi_k: U_k \rightarrow \mathbb R$ is an upper semicontinuous function such that

\begin{itemize}

\item $\phi_k$ is strictly plurisubharmonic

\item $\phi_k - \phi_{\ell}$ is pluriharmonic

\end{itemize}

\end{definition} 

If a K\"ahler cocycle $(U_k, \phi_k)_{k \in K}$ is smooth then we can associate to it the symplectic form $\omega$ which is $-dd^{\mathbb C}\phi_k$ on each $U_k$. Notice, for example, that a K\"ahler cocycle can consist of a single smooth plurisubharmonic function on all of $X$, as in the case of the smooth K\"ahler cocycle $((S^2 \backslash \{{\bf r_i}\})^{2n_1}, \widetilde{f}_i)$ on $(S^2 \backslash \{{\bf r_i}\})^{2n_1}$ and the singular K\"ahler cocycle $(M_i, \psi_i)$ on $M_i$.

\medskip

Perutz proves the following technical result.


\begin{lemma} \cite[Lemma 7.4]{MR2509747} Let $(U_k, \phi_k)$ be a continuous K\"ahler cocycle on a complex manifold $X$. Suppose that $X = X_1 \cup X_2$ such that $X_1$ and $X_2$ are open and the functions $\phi_k|_{U_k \cap X_1}$ are smooth. Then there exists a continuous function
%
\[
\chi: X \rightarrow \mathbb R, \text{ } Supp(\chi) \subset X_2
\]
%
\noindent and a locally finite refinement
%
\[
V_{\ell} \subset U_{k(\ell)} 
\]
%
\noindent such that the family $(V_{\ell}, \phi_{k(\ell)}|_{V_{\ell}} + \chi|_{V_{\ell}})$ is a smooth K\"ahler cocycle.

\end{lemma}

Notice that if $(U_k, \phi_k)$ happened to be the K\"ahler cocycle associated to a single $J$-convex function $\phi$ on $X$, then $(V_{\ell}, \phi_{\ell(k)}|_{V_{\ell}} + \chi|_{V_{\ell}})$ is the K\"ahler cocycle associated to the smooth plurisubharmonic function $\phi + \chi$. 

\medskip

In our case we take $X$ to be $\text{Sym}^{2n_1}(\Sigma(S) \backslash \{{\bf r_i}\})$, $X_1$ to be the complement of the main diagonal in this symmetric product, and $X_2$ to be a small neighborhood of the main diagonal with no intersection with $\mathbb T_{\widetilde{\boldsymbol \alpha}}$ and $\mathbb T_{\widetilde{\boldsymbol \beta}}$. Then the function $\psi_i: \text{Sym}^{2n_1}(S^2 \backslash \{{\bf r_i}\}) \rightarrow \mathbb R$ admits a smoothing to a $\text{Sym}^{2n_1}(j_i)$-convex function $\psi_i + \chi_i: \text{Sym}^{2n_1}(S^2 \backslash \{{\bf r_i}\}) \rightarrow \mathbb R$ which is equal to $\psi_i$ away from a neighborhood of the large diagonal. The symplectic form $\omega_{\phi_i+ \chi_i}$ is exact and compatible with the complex structure on $M$. 

\medskip

Finally, on $\mathbb T_{\widetilde{\boldsymbol \alpha}}$ the map $\chi_i$ is identically $0$ and $\psi_i = (2n_1)!\widetilde{f}_i|_{\alpha^1_1 \times \alpha^2_1 \times\cdots. \times \alpha^1_{n_1} \times \alpha^2_{n_1}}$. Therefore $\omega_{\psi_i+ \chi_i}|_{\mathbb T_{\widetilde{\boldsymbol \alpha}}} = 0$ and $d^{\mathbb C}(\psi_i + \chi_i)|_{\mathbb T_{\widetilde{\boldsymbol \alpha}}} = (2n_1-2)!d^{\mathbb C}(\widetilde{f_i})|_{\alpha^1_1 \times\cdots\times \alpha^2_{n_1}}$ is exact. Ergo $\mathbb T_{\widetilde{\boldsymbol \alpha}}$ is an exact Lagrangian in the exact symplectic manifold $M_i$, and similarly $\mathbb T_{\widetilde{\boldsymbol \beta}}$ is as well.

\medskip

The reader may at this point be alarmed that we have failed to check that $\tau$ is a symplectic involution.  We amend this by replacing $\omega_{\psi_i+\chi_i}$ by $\omega_i = \frac{1}{2}(\omega_{\psi_i+\chi_i} + \tau^*\omega_{\psi_i+\chi_i})$.  Since $\tau$ is a holomorphic involution, the exact form $\omega_i$ is $\text{Sym}^{2n_1}(j)$-compatible and nondegenerate.  Moreover, since $\tau$ preserves $\mathbb T_{\boldsymbol \alpha}$ and $\mathbb T_{\boldsymbol \beta}$, it is still the case that these two submanifolds are exact Lagrangians with respect to $\omega_i$.  This $\omega_i$ is our final choice of symplectic form on $M_i$.

\medskip

Let us now proceed to discuss the homotopy type and cohomology of $M_0^{\text{inv}}$, building up the structure we will use in the proof that this manifold satisfies the complex conditions which imply the existence of a stable normal trivialization.

\medskip

We claim $M_0^{\text{inv}}$ deformation retracts onto each of $\mathbb T_{\boldsymbol \alpha}$ and $\mathbb T_{\boldsymbol \beta}$.  To check this, we refer to a lemma whose proof is outlined in \cite[Lemma 5.1]{Hendricks} following an argument of Ong \cite{MR1993792}.

\begin{lemma}

The $r$th symmetric product of a wedge of $k$ circles $\vee_{i=1}^k S^1_i$ deformation retracts onto the $r$-skeleton of the $k$ torus $\prod_{i=1}^k S^1_i$, where each circle is given a CW structure consisting of the wedge point and a single one-cell, and the torus has the natural product CW structure.

\end{lemma}

We will apply this observation to $M_0^{\text{inv}}$. In $S^2$, let $\nu_{i}:[0,1] \rightarrow S^2$ be a small closed curve around $z_i$ for $0 \leq i \leq n_1$, such that $\nu_i$ is oriented counterclockwise in the complement of $w_{0}$. Then
%
%
\begin{align*}
H_1(S^2 \backslash \{{\bf z}\}) &= \mathbb Z\langle \nu_0, \nu_1,\cdots,\nu_{n_1} \rangle.
\end{align*}
%
Now $S^2\backslash \{{\bf w}\}$ deformation retracts onto a wedge of $n_1$ circles $\vee_{i=1}^{n_1} \nu'_i$, where $\nu_i'$ is a closed curve homotopic to $\nu_i$ which passes once through the origin.  Therefore $M_0^{\text{inv}} =\text{Sym}^{n_1}(S^2 \backslash \{{\bf w}\})$ deformation retracts onto the symmetric product of $\vee_{i=1}^{n_1} \nu'_i$, which in turn deformation retracts onto the product $\prod_{i=1}^{n_1} \nu'_i$.  However, this product is homotopy equivalent to $\prod_{i=1}^{n_1} \nu_i$, and indeed to $\prod_{i=1}^{n_1} \alpha_i$ and to $\prod_{i=1}^n \beta_i$.  We conclude that $M_0^{\text{inv}}$ has the homotopy type of an $n_1$-torus and in particular admits a deformation retraction onto each of $\mathbb T_{\boldsymbol \alpha}$ and $\mathbb T_{\boldsymbol \beta}$.

\medskip

We will require a concrete description of the cohomology rings of $M_0^{\text{inv}}$, $\mathbb T_{\boldsymbol \alpha}$, and $\mathbb T_{\boldsymbol \beta}$ for the computations of Section \ref{Stable Normal Triv Section}, so we pause to supply one now. Consider the one cycles
%
%
\begin{align*}
\overline{\nu_i}: [0,1] \rightarrow M^{\text{inv}} \\
t \mapsto (\nu_i(t)x_0\cdots x_0)
\end{align*}
%
\noindent where $x_0$ is any choice of basepoint. The $[\overline{\nu_i}]$ form a basis for $H_1(M^{\text{inv}})$.  Ergo, letting $\widehat{[\overline{\nu_i}]}$ denote the dual of $\overline{\nu_i}$, we have
%
%
%
\begin{align*}
H^1(M^{\text{inv}}) &= \mathbb Z \langle \widehat{[\overline{\nu_0}]},\cdots,\widehat{[\overline{\nu_{n_1}}]} \rangle \\
H^k(M^{\text{inv}}) &= \textstyle{\bigwedge^k}H^1(M^{\text{inv}})
\end{align*}


\medskip


Similarly, we can write down the homology of the tori $\mathbb T_{\boldsymbol \alpha}$ and $\mathbb T_{\boldsymbol \beta}$. Through a slight abuse of notation, let us insist that we have parametrizations $\alpha_i:[0,1] \rightarrow S^2$ running counterclockwise in $S^2 \backslash \{w_0\}$ and $\beta_i$ running clockwise. The first homology of $\mathbb T_{\boldsymbol \alpha}$ is generated by one-cycles
%
\begin{align*}
\overline{\alpha_i}: [0,1] &\rightarrow \alpha_1 \times \cdots \times \alpha_{n_1} \\
					t &\mapsto (y_1,\cdots,y_{i-1}, \alpha_i(t),y_{i+1},\cdots,y_{n_1}) \\					
\end{align*}
%
\noindent where $y_j = \alpha_j(0)$, and thus the cohomology of this torus is
%
%
\begin{align*}
H^1(\mathbb T_{\boldsymbol \alpha}) &= \mathbb Z\langle  \widehat{[\overline {\alpha_1}]},\cdots,\widehat{[\overline{ \alpha_{n_1}}]} \rangle \\
H^k(\mathbb T_{\boldsymbol \alpha}) &= \textstyle{\bigwedge^k}H^1(\mathbb T_{\boldsymbol \alpha})
\end{align*}
%
\noindent We apply analagous naming conventions to $\mathbb T_{\boldsymbol \beta}$, obtaining
%
\begin{align*}
H^1(\mathbb T_{\boldsymbol \beta}) &= \mathbb Z\langle  \widehat{[\overline {\beta_1}]},\cdots,\widehat{[\overline {\beta_{n_1}}]} \rangle \\
H^k(\mathbb T_{\boldsymbol \beta}) &= \textstyle{\bigwedge^k}H^1(\mathbb T_{\boldsymbol \beta})
\end{align*}
%
Moreover, observe that under the map on homology induced by inclusion $\iota: X \hookrightarrow M^{\text{inv}} \times[0,1]$, both $[\overline{\alpha_i}]$ and of $[\overline{\beta_i}]$ are sent to $[\overline{\nu_i}]$. Therefore the map $\iota^*$ on cohomology $H^k(M^{\text{inv}} \times [0,1]) \rightarrow H^k(X)$ is precisely the diagonal map, with
%
\[
\textstyle{\bigwedge_{j=1}^m}  \widehat{[\overline{\nu_{i_j}}]} \mapsto \textstyle{\bigwedge_{j=1}^m}  \widehat{[\overline{\alpha_{i_j}}]} + \textstyle{\bigwedge_{j=1}^m}  \widehat{[\overline{\beta_{i_j}}]}
\]


\begin{corollary}

The relative cohomology $H^*(M_0^{\text{inv}} \times [0,1], X)$ is the cohomology of the torus $(S^1)^{n_1}$, and in particular is torsion-free.

\end{corollary}

\begin{proof}

Consider the long exact sequence
%
\begin{equation*}
\xymatrix{
\cdots H^{m-1}(X) \ar[r] & H^m(M_0^{\text{inv}} \times[0,1], X)  \ar[r]^-{q^*} & H^m(M_0^{\text{inv}}) \ar[r]^-{\iota^*} &  H^m(X)\cdots
}
\end{equation*}

Taking into account that the obvious isomorphism between $H^*(\mathbb T_{\boldsymbol \alpha})$ and $H^*(\mathbb T_{\boldsymbol \beta})$ respects our labelling of the cohomology classes, $\iota^*$ is the diagonal map, and in particular an injection. When $m\geq 2$ we have the following short exact sequences.
%
\begin{equation*}
\xymatrix @R-.4cm{
0 \ar[r]& H^{m-1}(M_0^{\text{inv}} \times [0,1]) \ar[r]^-{\iota^*}\ar[r] &H^{m-1}(X) \ar[r] & H^m(M_0^{\text{inv}} \times [0,1], X) \ar[r] & 0 \\
& \textstyle{\bigwedge_{j=1}^{m-1}} \widehat{[\overline{\nu_{i_j}}]} \ar@{|->}[r] & \textstyle{\bigwedge_{j=1}^{m-1}} \widehat{[\overline{\alpha_{i_j}}]} + \textstyle{\bigwedge_{j=1}^{m-1}} \widehat{[\overline{\beta_{i_j}}]} & &
} 
\end{equation*}

\noindent Ergo for $m \geq 2$, $H^m(M_0^{\text{inv}} \times [0,1], X) \cong H^{m-1}(\mathbb T_{\boldsymbol \alpha}) \cong H^{m-1}(T_{\boldsymbol \beta})$.  (We will have occasion to be careful about the generators in Section \ref{Stable Normal Triv Section}.) The same result follows for $H^1(M_0^{\text{inv}} \times [0,1], X)$ trivially.




\end{proof}

Let us consider the implications of this result for the relative $K$-theory of $(M^{\text{inv}} \times [0,1], X)$, which is isomorphic to the reduced $K$-theory $\widetilde{K}((M^{\text{inv}} \times [0,1])/ X)$.  (For a review of $K$-theory, see \cite[Chapter 2]{HatcherKTheory}; all of the facts used in this paper are also summarized in \cite[Section 6]{Hendricks}.) Notice that $(M^{\text{inv}} \times [0,1], X)$ deformation retracts onto the compact CW pair $(\text{Sym}^{n_1}(S^2 \backslash \cup_i \nu(w_i)) \times [0,1], X)$, where $\nu(w_i)$ is a small open neighborhood.  This deformation retraction makes it legitimate to consider the reduced $K$-theory of the pair by identifying it with $\widetilde{K}((\text{Sym}^{n_1}(S^2 \backslash \cup_i \nu(w_i)) \times [0,1])/ X)$.  From now on we will apply this trick without mention.

\medskip

Recall that there is a rational ring isomorphism
%
\begin{align*}
\widetilde{\text{ch}}:(\widetilde{K}^0(X) \oplus \widetilde{K}^1(X))\otimes \mathbb Q \rightarrow \widetilde{H}^*(X; \mathbb Q)
\end{align*}
%
\noindent between the rational reduced $K$-theory and rational reduced cohomology of any space $X$ chosen such that if $V$ is a line bundle over $X$ and $c_1(V)$ is the first Chern class of $X$, $\widetilde{\text{ch}}(V) = \sum_{i=1}^{\infty} \frac{c_1(V)^i}{i!}$ and $\widetilde{\text{ch}}$ is a ring homomorphism.  Using this isomorphism and the Atiyah-Hirzebruch spectral sequence, Atiyah and Hirzebruch have shown that if the reduced cohomology $\widetilde{H}^*(X)$ is torsion-free, the reduced $K$-theory $\widetilde{K}^*(X)$ is as well, and the stable isomorphism class of a vector bundle is determined entirely by its Chern classes \cite[Section 2.5]{MR0139181}.  In particular, since $H^*(M^{\text{inv}} \times [0,1], X) = \widetilde{H}^*((M^{\text{inv}} \times [0,1])/ X)$ is torsion-free, the stable isomorphism class of a complex vector bundle over $(M_0^{\text{inv}}, X)$ -- that is, a bundle whose restriction to $X$ is stably trivial -- is entirely determined by its Chern classes.  To show such a bundle is stably trivial, it suffices to show that all its Chern classes are zero.




\section{Stable Normal Triviality of the Normal Bundle} \label{Stable Normal Triv Section}

Having seen that the symplectic geometry conditions of Seidel and Smith's theory are satisfied for $(M_i, L_0, L_1)$, when $i=0,1,2$, we now proceed to check that $(M_0, L_0, L_1)$ fulfills the complex conditions that, by Lemma \ref{Nullhomotopy Lemma}, imply the existence of a stable normal trivialization.

\begin{proposition} \label{Stable Normal Trivialization}

Consider the complex manifold $M_0=\text{Sym}^{2n_1}(S^2 \backslash \{\widetilde{{\bf w}}\})$ together with its totally real submanifolds $L_0 = \mathbb T_{\boldsymbol \beta}$ and $L_1= \mathbb T_{\boldsymbol \alpha}$, and the holomorphic involution $\tau$ which preverves $L_0$ and $L_1$.  The map
%
\begin{align*}
(M_0^{\text{inv}} \times [0,1], (L_0 \times \{0\}) \cup (L_1 \times \{1\})) \rightarrow (BU, BO)
\end{align*}
%
\noindent which classifies the pullback $\Upsilon(M_0^{\text{inv}}) = N(M_0^{\text{inv}}) \times [0,1]$ of the complex normal bundle of $M_0^{\text{inv}}$ together with the totally real subbundles $NL_0^{\text{inv}} \times\{0\}$ over $L_0^{\text{inv}} \times \{0\}$ and $J(NL_1^{\text{inv}}) \times \{1\}$ over $L_1^{\text{inv}} \times \{1\}$ is nulhomotopic.


\end{proposition}


As a first step, we must establish the complex triviality of $NM_0^{\text{inv}}$ (and thus of $\Upsilon(M_0^{\text{inv}}$)) and the real triviality of $NL_i^{\text{inv}}$ for $i=1,2$.

\begin{lemma}

The complex bundle $NM_0^{\text{inv}}$ is stably trivial.

\end{lemma}

\begin{proof}

The inclusion map $\iota_1: (S^2 \backslash \{{\bf w}\}) \hookrightarrow S^2$ is nulhomotopic.  Therefore the induced inclusion $\text{Sym}^{n_1}(\iota_1):\text{Sym}^{n_1}(S^2 \backslash 
\{{\bf w}\}) \hookrightarrow  \text{Sym}^{n_1}(S^2)$ is also nulhomotopic. The normal bundle of $M_0^{\text{inv}} = \text{Sym}^{n_1}(S^2 \backslash \{{\bf w}\})$ in $M_0 = \text{Sym}^{2n_1}(S^2 \backslash \{{\bf w}\})$ is exactly the restriction of the normal bundle 
to $\text{Sym}^{n_1}(S^2)$ in $\text{Sym}^{2n_1}(S^2)$ along the inclusion map $\text{Sym}^{n_1}(\iota_1)$.  As the map $\text{Sym}^{n_1}(\iota_1)$ is nulhomotopic, $NM_0^{\text{inv}}$ is stably trivial.\end{proof}


The proof of the next lemma proceeds exactly as in \cite[Lemma 7.3]{Hendricks}.

\begin{lemma} \label{Torus Triviality Lemma}
The normal bundles of $\mathbb T_{\boldsymbol \alpha} \subset \mathbb T_{\widetilde{\boldsymbol \alpha}}$ and $\mathbb T_{\boldsymbol \beta} \subset \mathbb T_{\widetilde{\boldsymbol \beta}}$ are trivial.
\end{lemma}



\medskip

We now turn to the question of relative triviality.  Let $X = (L_0 \times \{0\}) \cup (L_1 \times \{1\}))$ as in Section \ref{Geometry Section}.  Choose preferred trivializations of the totally real bundles  $NL_0^{\text{inv}} \times \{0\}$ and $J(NL_1^{\text{inv}}) \times \{1\}$ and tensor with $\mathbb C$ to extend to a preferred trivialization of the complex bundle $\Upsilon(M_0^{\text{inv}})|_{X}$. We use this trivialization to pull back $[\Upsilon(M_0^{\text{inv}})] \in \widetilde{K}^0(M_0^{\text{inv}} \times [0,1])$ to a relative bundle $[\Upsilon(M_0^{\text{inv}})]_{\text{rel}} \in \tilde{K}^0((M_0^{\text{inv}} \times [0,1])/X)$. Because the reduced cohomology, and therefore the reduced $K$-theory, of $(M_0^{\text{inv}}\times [0,1])/X$ have no torsion, to verify triviality of the relative bundle, it suffices to show that the Chern classes of $[\Upsilon(M_0^{\text{inv}})]_{\text{rel}}$ are trivial. 

\begin{remark} 
It may be helpful to draw attention to a minor problem of notation: in earlier sections, $\widetilde{K}$ is a doubly periodic knot, but also in the present section $\widetilde{K}^0(B)$ is the reduced $K$-theory of the topological space $B$.  We hope this will not occasion confusion.
\end{remark}

Fundamentally, the argument for relative triviality rests on the fact that each of the $n_1$ linearly independent periodic domains in $S^2 \backslash \{{\bf w}\}$ has Maslov index zero. Therefore, we pause here to recall the notation of Section \ref{Periodic Knots Section}. Let $x_i$ be the single positive intersection point in $\alpha_i \cap \beta_i$ and $y_i$ the negative intersection point.  Let $F_i$ be the closure of the component of $S - \alpha_i - \beta_i$ containing $z_{i}$ and $E_i$ be the closure of the component of $S - \alpha_i -\beta_i$ containing $z_{i+1}$ (or $z_1$ if $i=n_1$).  Then $P_i = E_i - F_i$ is a periodic domain of index zero on $\mathcal D$ with boundary $\beta_i - \alpha_i$.  Finally, let $\gamma_i$ be the union of the arc of $\alpha_i$ running from $x_i$ to $y_i$ and the arc of $\beta_i$ running from $x_i$ to $y_i$.  In particular, this specifies that $\gamma_i$ has no intersection with any $\alpha$ or $\beta$ curves other than $\alpha_i$ and $\beta_i$, and moreover the component of $S - \gamma_i$ which does not contain $w_0$ contains only a single basepoint $w_i$. See Figure \ref{Periodic Domains Figure} for an illustration of the domain $P_i$.


\begin{figure}
\caption{The periodic domain $P_i = E_i - F_i$ has Maslov index zero.}
\label{Periodic Domains Figure}
\begin{tikzpicture}
\def \betacurve {(1.25,0)ellipse (54 pt and 24 pt)};
\def \alphacurve {(-1.25,0)ellipse (54 pt and 24 pt)};
\node(1)[font = \small] at (0,0) {${w_i}$};
\node(2)[font=\small] at (-2.5,0){${z_i}$};
\node(3)[font=\small] at (2.5,0){${z_{i+1}}$};
\node(4) at (-1.5,0){$F_i$};
\node(5) at (1.5,0){$E_i$};
\node(6) at (-2,1.1){$\alpha_i$};
\node(7) at (2,1.1){$\beta_i$};
\node(9) at (0,1)[font = \small]{$x_i$};
\draw [red][line width = 2 pt] (-1.25,0)ellipse (55 pt and 25 pt);
\draw [blue, dashed][line width = 2 pt](1.25,0)ellipse (55 pt and 25 pt);
\begin{scope}[even odd rule]
\clip \alphacurve \betacurve;
\fill[gray, opacity = .5] \betacurve;
\fill[gray, opacity = .5] \alphacurve;
\end{scope}
\end{tikzpicture}
\end{figure}

\medskip

The structure of the argument is as follows: for each even $k$ such that $1 \leq k \leq 2n_1$, we will use the periodic domains $P_i$ to produce a set of $k$-chains $\{W_{\bf I}\}$ in $(M_0^{\text{inv}} \times [0,1], X)$ whose relative homology classes generate the $k$th homology of $H_k(M_0^{\text{inv}} \times[0,1], X)$.  We will then show that the restriction of $\Upsilon(M^{\text{inv}}_0)|_{\text{rel}}$ to each $W_{\bf I}$ is trivial as a relative vector bundle, and that therefore $\langle c_k(\Upsilon(M_0^{\text{inv}})|_{\text{rel}}), [W_{\bf I}] \rangle=0$.  Since the $[W_{\bf I}]$ generate $H_k(M_0^{\text{inv}}\times[0,1], X)$, we will have proven that $c_k(\Upsilon(M_0^{\text{inv}})|_{\text{rel}})$ is identically zero.

\medskip

More specifically, we will describe the chains $W_{\bf I}$ as a subset of a product of two-chains $Y_i$ in $(S^2 \backslash \{{\bf w}\}) \times [0,1]$ such that the projection to $S^2 \backslash \{{\bf w}\}$ is the periodic domain $P_i$ and the $Y_i$ are pairwise disjoint. We will then show that the restriction of the bundle $\Upsilon(M^{\text{inv}}_0)|_{\text{rel}}$ to $W_{\bf I}$ also breaks up as the restriction of a product of relative bundles $\Upsilon(Y_i)$ over the $Y_i$ which are known to be trivial through Maslov index arguments.  Let us begin by constructing these manifolds $Y_i$.

\medskip

For $1 \leq i \leq n_1$, let $Y_i$ be a subspace of $S^2 \times [0,1]$ with the following properties: 

\medskip

\begin{itemize}

\item $Y_i$ is topologically $S^1 \times [0,1]$, and $Y_i \cap (S^2 \times \{t\})$ is $S^1$ for all $t$.

\item The boundary of $Y_i$ is $\beta_i \times \{0\} - \alpha_i \times \{1\}$.

\item The projection of $Y_i$ to the punctured sphere is a copy of the periodic domain $P_i$.

\item $Y_i \cap Y_j = \emptyset$ if $i \neq j$.

\item If $x_i$ is the point of positive intersection of $\alpha_i$ and $\beta_i$, $\{x_i\} \times [0,1] \subset Y_i$.

\end{itemize}

\medskip

The 2-chains $Y_i$ are constructed as follows: for $t \in [0, \frac{1}{2}]$, choose a linear homotopy $H_t$ from $\beta_i$ to $\gamma_i$ inside the domain $E_i$ which fixes $\beta_i \cap \gamma_i$. Let the intersection of $Y_i$ with $S^2 \times \{t\}$ be the embedded circle $H_t(\beta_i) \times \{t\}$. Similarly, for $t \in [\frac{1}{2}, 1]$, choose a linear homotopy $J_t$ from $\gamma_i$ to $\alpha_i$ inside the domain $F_i$ which fixes $\alpha_i \cap \gamma_i$, and let the intersection of $Y_i$ with $S^2 \times \{t\}$ be $J_t(\gamma_i)$.

\medskip

Observe that this description of $Y_i$ has the following properties: first, $Y_i$ is contained in $(S^2 \backslash \{{\bf w}\}) \times [0,1]$ as promised.  Second, $Y_i$ contains the line segment $\{x_i\} \times [0,1]$.  Third, the intersection of $Y_i$ with $(\alpha_i \cup \beta_i)\times (0,1)$ is entirely contained in the cylinder $\gamma_i \times (0,1)$.  Finally, the projection of $Y_i \cap (S^2 \times [0, \frac{1}{2}])$ to $S^2$ lies entirely ``inside'' $\beta_i$ - that is, on the component of $S^2 \backslash \beta_i$ not containing $w_0$ - implying that the sets $Y_i \cap (S^2 \times [0, \frac{1}{2}])$ are pairwise disjoint.  Similarly, the projection of $Y_i \cap (S^2 \times [\frac{1}{2},1])$ to $S^2$ lies entirely ``inside'' $\alpha_i$, and therefore the sets  $Y_i \cap (S^2 \times [\frac{1}{2},1])$ are pairwise disjoint.  Ergo the $Y_i$ are pairwise disjoint.

\medskip

We are now ready to define the complex line bundles $\Upsilon(Y_i)$.  Recall from Section \ref{Floer Cohomology Section} that there is a holomorphic embedding
%
\begin{align*}
\iota :M^{\text{inv}}_0 = \text{Sym}^{n_1}(S^2 \backslash \{{\bf w}\}) &\rightarrow \text{Sym}^{n_1}(S^2 \backslash \{\widetilde{\bf w}\}) = M_0 \\
(x_1...x_{n_1}) &\mapsto (x_1^1x_1^2...x_{n_1}^1 x_{n_1}^2).
\end{align*}
%
\noindent Similarly, we have a holomorphic embedding
%
\begin{align*}
\iota': (S^2 \backslash \{{\bf w}\}) &\rightarrow \text{Sym}^2(S^2 \backslash \{\widetilde{\bf w}\}) \\
x &\mapsto (x^1x^2)
\end{align*}
%
which takes a point $x$ on $S^2 \backslash \{{\bf w}\}$ to the unordered pair consisting of its two (not necessarily distinct) lifts on $S^2 \backslash \{\widetilde{{\bf w}}\}$ under the projection map $\pi: (S^2 \backslash \{\widetilde{\bf w}\}) \rightarrow (S^2 \backslash \{{\bf w}\})$. Notice that for each $1 \leq i \leq n_1$, $\iota'(\alpha_i) \subset \alpha_i^1 \times \alpha_i^2$ and $\iota'(\beta_i) \subset \beta_i^1 \times \beta_i^2$.  Let $N(S^2 \backslash \{{\bf w}\})$ be the normal bundle to $S^2 \backslash \{{\bf w}\}$ in the second symmetric product $\text{Sym}^2(S^2 \backslash \{\widetilde{\bf w}\})$.  This is a complex line bundle over a punctured sphere, hence trivial.  For each $1 \leq i \leq n_1$, let $N\alpha_i$ be the real normal bundle to $\alpha_i$ in $\alpha_i^1 \times \alpha_i^2$ and $N\beta_i$ be the real normal bundle to $\beta_i$ in $\beta_i^1 \times \beta_i^2$.  An argument similar to that of Lemma \ref{Torus Triviality Lemma} shows that $N\alpha_i$ and $N\beta_i$ are trivial real line bundles for all $i$.

\medskip

Consider the pullback of the normal bundle $N(S^2 \backslash \{{\bf w}\})$ to $(S^2 \backslash \{{\bf w}\}) \times [0,1]$. The complex line bundle over $Y_i$ that interests us is the restriction of this pullback to $Y_i$, which we shall by analogy denote $\Upsilon(Y_i)$.  That is, $\Upsilon(Y_i) = (N(S^2 \backslash \{{\bf w}\}) \times [0,1])|_{Y_i}$. This complex line bundle has totally real subbundles $J(N\alpha_i \times \{1\})$ over $\alpha_i \times \{1\}$ and $N\beta_i \times \{0\}$ over $\beta_i \times  \{0\}$. Choose preferred real trivializations of these real line bundles, and extend to a complex trivialization of $\Upsilon(Y_i)|_{(\beta_i \times \{0\}) \cup (\alpha_i \times \{1\})}$ by tensoring with $\mathbb C$. For convenience, let this subspace $(\beta_i \times \{0\}) \cup (\alpha_i \times \{1\})$ be $X_i$.

\begin{lemma}

For $1 \leq i \leq n_1$, given any preferred trivialization of $\Upsilon(Y_i)|_{X_i}$ as a complex vector bundle, the relative vector bundle $\Upsilon(Y_i)|_{\text{rel}}$ over $(Y_i,X_i)$ is stably trivial.

\end{lemma}

\begin{proof}

Let $\iota_i$ be the inclusion of $Y_i$ into $(S^2 \backslash \{{\bf w}\})\times[0,1]$, and $p$ be the projection of $(S^2 \backslash \{{\bf w}\})\times [0,1]$ to  $S^2 \backslash \{{\bf w}\}$. Then the image of $p \circ \iota_i(Y_i)$ is the periodic domain $P_i$.  

\medskip

Consider the commutative diagram below. The top horizontal inclusion of $S^2 \backslash \{w\}$ into $\text{Sym}^{n_1}(S^2 \backslash \{{\bf w}\})$ is defined by mapping a point $x$ to $(x x_1...\widehat{x_i}...x_{n_1})$, where again each $x_j$ is the positively oriented point in $\alpha_j \cup \beta_j$.  The bottom inclusion of $\text{Sym}^2(S^2 \backslash \{\widetilde{\bf w}\})$ into $\text{Sym}^{2n_1}(S^2 \backslash \{\widetilde{\bf w}\})$ sends an unordered pair $(xy)$ to $(xyx_i^1x_i^2...\widehat{x_i^1}\widehat{x_i^2}...x_{n_1}^1x_{n_1}^2)$. 


\[
\xymatrix{
Y_i \ar[r]^-{p \circ \iota_i} \ar[dr] & S^2 \backslash \{{\bf w}\} \ar@{^{(}->}[d]^-{\iota'} \lhook\mkern-7mu\ar[r]& \text{Sym}^{n_1}(S^2 \backslash \{{\bf w}\}) \ar@{^{(}->}[d]^-{\iota} \\
& \text{Sym}^2(S^2 \backslash \{\widetilde{\bf w}\}) \lhook\mkern-7mu\ar[r]& \text{Sym}^{2n_1}(S^2 \backslash \{\widetilde{\bf w}\})
}
\]

Consider the map $\phi:Y_i \rightarrow \text{Sym}^{n_1}(S^2 \backslash \{{\bf w}\})$ given by composition along the top row of the diagram.  This is a topological annulus representing the periodic domain $P_i$. Since $P_i$ has Maslov index zero, by the discussion in Section \ref{Heegaard Floer Background Section}, the first Chern class of the pullback of the complex tangent bundle $T\text{Sym}^{n_1}(S^2 \backslash \{{\bf w}\})$ to $Y_i$ relative to the complexification of the pullbacks of the real tangent bundles $J(T(\mathbb T_{\boldsymbol \alpha}))$ to one component of the boundary of $Y_i$ and $T(\mathbb T_{\boldsymbol \beta})$ to the other is zero. However, the pullback $\phi^*(T\text{Sym}^{n_1}(S^2 \backslash \{{\bf w}\}))$ of the tangent bundle of the total symmetric product to $Y_i$ is exactly $(p \circ \iota)_*(T(S^2 \backslash \{{\bf w}\}) \oplus \mathbb C^{n_1 - 1})$, where the factors of $\mathbb C$ are the restriction of the tangent bundle of the punctured sphere to the points $x_j$ such that $j \neq i$. Moreover, $(p \circ \iota)^*(T(S^2 \backslash \{{\bf w}\}))$ is precisely the restriction of the pullback bundle $p^*(T(S^2 \backslash \{{\bf w}\})) = T((S^2 \backslash \{{\bf w}\})\times [0,1])$ over $(S^2 \backslash \{{\bf w}\}) \times [0,1]$ to the subspace $Y_i$.  Therefore $\phi^*(T\text{Sym}^{n_1}(S^2 \backslash \{{\bf w}\})) = T((S \backslash \{{\bf w}\}) \times [0,1])|_{Y_i} \oplus \mathbb C^{n_1 - 1}$.

\medskip

Similarly, the pullback $\phi^*(J(T (\mathbb T_{\boldsymbol \alpha})))$ to the boundary component $\alpha_i \times \{1\} \subset Y_i$ is $(p \circ \iota_i)^*(J(T(\alpha_i)\oplus \mathbb R^{n_1-1}))$, where the factors of $\mathbb R$ are the canonical real subspace of the tangent bundle to the punctured sphere at the points $x_j$ for $j \neq i$.  The pullback of this bundle to $\alpha_i \times \{1\} \subset Y_i$ is $J((T(\alpha_i) \times \{1\}) \oplus \mathbb R^{n_1-1})$.  A similar argument shows that $\phi^*(J(T(\mathbb T_{\boldsymbol \beta})))$ is $(T(\beta_i)\times \{0\}) \oplus \mathbb R^{n_1-1}$.  Therefore we have seen that the complex vector bundle $T((S^2 \backslash \{{\bf w}\})\times [0,1])|_{Y_i} \oplus \mathbb C^{n_1-1}$ relative to the complexification of its totally real subbundles $(T(\beta_i)\times \{0\})\oplus \mathbb R^{n_1+1}$ over $\beta_i \times \{0\}$ and $J((T(\alpha_i)\times \{1\})\oplus \mathbb R^{n_1+1})$ over $\alpha_i \times \{1\}$ has relative first Chern class zero. Hence the same is true of the complex line bundle $T((S^2 \backslash \{{\bf w}\})\times [0,1])|_{Y_i}$ relative to the complexification of its totally real subbundles $J(T(\alpha_i)\times \{1\})$ and $T(\beta_i)\times\{0\}$.  As this is a line bundle, triviality of the first relative Chern class suffices to show stable triviality of the relative vector bundle.

\medskip

Now consider the map $\widetilde{\phi} = \iota \circ \phi$ from $Y_i$ to $\text{Sym}^{2n_1}(S^2 \backslash \{\widetilde{\bf w}\})$ given by any path through the diagram from $Y_i$ to $\text{Sym}^{2n_1}(S^2 \backslash \{\widetilde{\bf w}\})$.  This is a topological annulus representing the periodic domain $\pi^{-1}(P_i)$ in $\widetilde{\mathcal D}$, which also has Maslov index zero.  Therefore the pullback along $\widetilde{\phi}$ of the complex tangent bundle $T(\text{Sym}^{2n_1}(S^2 \backslash \{\widetilde{\bf w}\}))$ to $Y_i$ relative to complexifications of pullbacks of the totally real subbundles $J(T(\mathbb T_{\widetilde{\boldsymbol \alpha}}))$ and $T(\mathbb T_{\widetilde{\boldsymbol \beta}})$ to $\alpha_i \times \{1\}$ and $\beta_i \times \{0\}$ has trivial relative first Chern class. However, once again the pullback of this relative bundle along the inclusion map $\text{Sym}^{2}(S^2 \backslash \{\widetilde{\bf w}\}) \hookrightarrow \text{Sym}^{2n_1}(S^2 \backslash \{\widetilde{\bf w}\})$ decomposes into a much smaller tangent bundle together with a trivial summand. Observe that $\widetilde{\phi}^*(T(\text{Sym}^{2n_1}(S^2 \backslash \{\widetilde{\bf w}\}))$ is $(\iota' \circ (p \circ \iota_i))^*(T(\text{Sym}^{2}(S^2 \backslash \{\widetilde{\bf w}\})) \oplus \mathbb C^{2n_1 - 2}))$.  Pulling back along the inclusion map $\iota'$, we see that this bundle breaks up as $T(S^2 \backslash \{{\bf w}\}) \oplus N(S^2 \backslash \{{\bf w}\}) \oplus \mathbb C^{2n_1 - 2}$ over $S^2 \backslash \{{\bf w}\}$, and that therefore its ultimate pullback to $Y_i$ is $T((S^2 \backslash \{{\bf w}\}) \times [0,1]) \oplus \Upsilon(Y_i) \oplus \mathbb C^{2n_1 -2}$. 

\medskip

Similarly, the pullback along $\widetilde{\phi}$ of $J(T(\mathbb T_{\widetilde{\boldsymbol \alpha}}))$ to $\alpha_i \times \{1\}$ is precisely $J(T{\alpha_i} \times \{1\}) \oplus J(N{\alpha_i} \times \{1\}) \oplus J(\mathbb R^{2n_1-2})$.  Finally, the pullback along $\widetilde{\phi}$ of $T(\mathbb T_{\widetilde{\boldsymbol \beta}})$ to $\beta_i \times \{0\}$ is $(T{\beta_i} \times \{0\}) \oplus (N_{\beta_i} \times \{0\}) \oplus \mathbb R^{2n_1-2}$.  Dropping the trivial summands, we conclude that the first relative Chern class of $T((S^2 \backslash \{{\bf w}\}) \times [0,1]) \oplus \Upsilon(Y_i)$ relative to complexifications of the totally real subbundle $(T{\beta_i} \times \{0\}) \oplus (N_{\beta_i} \times \{0\})$ over $\beta_i \times \{0\}$ and the totally real subbundle $J(T{\alpha_i} \times \{1\}) \oplus J(N{\alpha_i} \times \{1\})$ over $\alpha_i \times \{1\}$ has relative first Chern class zero.  This, combined with our previous conclusions concerning relative triviality of $T((S^2 \backslash \{{\bf w}\}) \times [0,1])$ with respect to its subbundles $T\beta_i \times \{1\}$ over $\beta_i \times \{1\}$ and $J(T\alpha_i \times \{0\})$ over $\alpha_i \times \{0\}$, implies that $\Upsilon(Y_i)$ has relative first Chern class zero with respect to complexifications of its subbundles $N\beta_i \times \{0\}$ and $J(N\alpha_1 \times \{1\})$, as promised.


\end{proof}


We are now ready to construct generators for $H_k(M_0^{\text{inv}} \times [0,1], X)$, and use them to prove that the relative Chern classes of $[\Upsilon(M_0^{\text{inv}})]$ are identically zero.  Recall that for $k > 1$ there are short exact sequences on homology
%
\[
\xymatrix{
0 \ar[r] & H_{k+1}(M_0^{\text{inv}} \times [0,1], X) \ar[r] & H_k(\mathbb T_{\boldsymbol \beta}) \oplus H_k(\mathbb T_{\boldsymbol \alpha}) \ar[r]^-{\iota_*} & H_k(M_0^{\text{inv}}) \ar[r] &0.
}
\]

As in Section \ref{Geometry Section}, for $k>1$, the $k+1$st homology group $H_{k+1}(M_0^{\text{inv}} \times [0,1], X)$ is the kernel of the map $(\iota_*)$.  Let us take a closer look at generators for this group. We have seen that the kernel of $\iota_*$ is $\mathbb Z \langle \textstyle{\bigwedge_{j=1}^{k}} [\beta_{i_j}] \oplus -\textstyle{\bigwedge_{j=1}^k} [\alpha_{i_j}] : 1 \leq i_1 < \cdots <i_k \leq n_1 \rangle$. Therefore for each ${\bf I} = (i_1,...,i_k)$ such that $1 \leq i_1 < \cdots < i_k \leq n_1$, there is a $(k+1)$-chain $W_{\bf I}$ in $M_0^{\text{inv}} \times [0,1]$ whose boundary is $\prod_{j=1}^{k} \beta_{i_j} - \prod_{j=1}^k \alpha_{i_j}$. We will describe this chain in terms of the two-chains $Y_i$.

\medskip

Construct the $k$ chain $W_{\bf I}$ as follows. Let $W_{\bf I}$ be topologically $(S^1)^k \times [0,1]$, and insist that the intersection of $W_{\bf I}$ with $M_0^{\text{inv}} \times \{t\}$ is the product $\prod_{j=1}^k Y_{i_j} \cap (S^2 \backslash \{{\bf w}\}) \times \prod_{i \notin {\bf I}}\{x_i\}$. Since the intersection of each $Y_{i_j}$ with $(S^2 \backslash \{{\bf w}\}) \times \{t\}$ is a circle for each $t$, this says that the intersection of $W_{\bf I}$ with $M_0^{\text{inv}}\times \{t\}$ is a $k$-torus for all $t$.

\medskip

Notice that for all $t$, $W_{\bf I} \cap (M_0^{\text{inv}} \times \{t\})$ is a subset of the product $\prod_{i=1}^{n} (Y_i \cap ((S^2 \{{\bf w}\})\times [0,1])$.  (Recall here that the line segments $\{x_i\} \times [0,1]$ are subsets of $Y_i$ for each $i$.) Since the $Y_i$ are pairwise disjoint, each $W_{\bf I}\cap(M_0^{\text{inv}}\times \{t\})$ is a submanifold of $M_0^{\text{inv}} = \text{Sym}^{n_1}(S^2 \backslash \{{\bf w}\})$ and is disjoint from the fat diagonal.  Furthermore, since $W_{\bf I}$ is homeomorphic to $(S^1)^k \times [0,1]$ and $\partial W_{\bf I}$ is 
$\prod_{j=1}^{k} \beta_{i_j} - \prod_{j=1}^k \alpha_{i_j}$, we see that the collection $\{W_{\bf I}: 1 \leq i_1 < \cdots < i_k \leq n_1\}$ generates $H^k(M_0^{\text{inv}}\times [0,1], X)$.

\medskip

Let us consider the restriction of $\Upsilon(M_0^{\text{inv}})$ to $W_{\bf I}$ for some particular ${\bf I}$.  We pick preferred trivializations of $N(L_0) \times \{0\}) = N(\mathbb T_{\boldsymbol \beta}) \times \{0\}$ and $J(N(L_1)  \times \{1\}) = J(N(\mathbb T_{\boldsymbol \alpha}) \times \{1\})$ which are products of preferred trivializations of the real subbundles $N(\beta_i) \times \{0\}$ and $J(N(\alpha_i) \times \{1\})$ in $\Upsilon(Y_i)$.

\medskip

Because $W_i \cap (M_0^{\text{inv}} \times \{t\})$ lies entirely off the fat diagonal for each $t$, the restriction of $\Upsilon(M_0^{\text{inv}})$ to each $W_i$ also decomposes as a product bundle.  In other words, regard $W_i$ as a subset of the abstract product $\prod_{i=1}^{n_1} Y_i$ (which is \textit{not} a submanifold of $M_0^{\text{inv}} \times [0,1]$). Then $\Upsilon(M_0^{\text{inv}})|_{W_{\bf I}}$ is the restriction of the product bundle $(\prod\Upsilon(Y_i))|_{W_{\bf I}}$.  Therefore, since each $\Upsilon(Y_i)$ admits a trivialization with respect to a choice of trivialization of the bundle restricted to $\alpha_i$ and $\beta_i$, we conclude that $\Upsilon(M_0^{\text{inv}})|_{W_{\bf I}}$ is trivializable with respect to preferred trivializations of the bundle over $\mathbb T_{\boldsymbol \beta}\times \{0\}$ and $J(\mathbb T_{\boldsymbol \alpha} \times \{1\})$.


\begin{remark}
We could as easily have shown that $(\text{Sym}^{2n_1}(S^2 \backslash \{{\bf z}\}), \mathbb T_{\widetilde{\boldsymbol \beta}}, T_{\widetilde{\boldsymbol \alpha}})$ has a stable normal trivialization. In the case studied here this would produce a spectral sequence from $\widehat{\mathit{HF}}(S^3) \otimes W^{\otimes 2n_1}$ to $\widehat{\mathit{HF}}(S^3) \otimes W^{\otimes n_1}$, which is not especially interesting.
\end{remark}


\section{Examples}

In this section we produce some examples of the behavior of the spectral sequences of Theorems \ref{Link Floer Homology Spectral Sequence} and \ref{Knot Floer Homology Spectral Sequence}.  We present the cases of the unknot and the trefoil as doubly-periodic knots. To simplify matters, we will compute all spectral sequences using appropriate chain complexes tensored with $\mathbb Z_2((\theta))$, instead of tensoring with $\mathbb Z_2[[\theta]]$ and later further tensoring the $E^{\infty}$ page with $\theta^{-1}$. Then the $E^1$ page of each spectral sequence is a free module with generators the link Floer or knot Floer homology of our doubly periodic knot, and the $E^{\infty}$ page is a free module over $\mathbb Z((q))$ with generators the link or knot Floer homology of the quotient.  (The only information we lose by this change is that the $E^{\infty}$ page is not the borel Heegaard Floer link or knot homology, but rather the borel homology tensored with $\theta^{-1}$.)

\subsection{The case of the Unknot}

In the simplest possible case, let $\widetilde{K}$ be an unknot, and $K$ its quotient knot, a second unknot.  Consider the diagram $\mathcal D$ for $K \cup U$ on the sphere $S^2 = \mathbb C^2$ in which $K$ is the unit circle with basepoints $w_0, z_0$ on $U$ and basepoints $w_1, z_1$ on $K$ such that $w_0$ lies at $\infty$, $z_0$ lies at $0$, $w_1$ lies at $i$, and $z_1$ lies at $-i$.  Supply a single $\alpha_1$ and $\beta_1$ as in Figure \ref{Unknot Heegaard Diagrams Figure} coherently with a clockwise orientation of $K$.

\begin{figure} 
\caption{An equivariant Heegaard diagram for the unknot together with an axis (i.e. a Hopf link), and its quotient Heegaard diagram (another Hopf link).}
\label{Unknot Heegaard Diagrams Figure}
\subfloat{\begin{tikzpicture}[scale=.9]
\centering
\tikzstyle{every node}=[font=\small];
\node(1) at (0,1.8){$w_1^1$};
\node(2) at (0,-1.8){$w_1^2$};
\node(3) at (1.8,0){$z_1^1$};
\node(4) at (-1.8,0){$z_1^2$};
\node(5) at (0,0){$z_0$};
\node(6) at (-2.5,2.5){$w_0$};
\tikzstyle{every node}=[font=\tiny];
\node(7) at (0,2.6){$a^1$};
\node(8) at (.05,.65){$b^1$};
\node(9) at (0,-2.6){$a^2$};
\node(10) at (.05,-.65){$b^2$};
\node(11) at (2.6,0){$e^1$};
\node(12) at (.65,0){$c^1$};
\node(13) at (-2.6,0){$e^2$};
\node(14) at (-.65,0){$c^2$};
\tikzset{every path/.style={line width = 2 pt}};
\draw [rotate=-45][red](0,1.4) ellipse (49pt and 20pt);
\draw[rotate=-45][magenta](0,-1.4) ellipse (49pt and 20pt);
\draw[rotate=45][dashed][blue](0,1.4) ellipse (49pt and 20pt);
\draw[rotate=45][dashed][cyan](0,-1.4) ellipse (49pt and 20pt);
\end{tikzpicture}}
\hspace{1 cm}
\subfloat{\begin{tikzpicture}[scale = .9]
\centering
\tikzstyle{every node}=[font=\small];
\node(1) at (0,0){$z_0$};
\node(2) at (-2.5,2.5){$w_0$};
\node(3) at (0,1.75){$w_1$};
\node(4) at (0,-1.75){$z_1$};
\tikzstyle{every node}=[font=\tiny];
\node(5) at (0,2.5){$a$};
\node(6) at (0,1){$b$};
\node(7) at (0,-1){$c$};
\node(8) at (0,-2.5){$e$};
\tikzset{every path/.style={line width = 2 pt}};
\draw[blue][dashed](0,2.25)..controls (-3, 2.25) and (-3,-2.25)..(0,-2.25);
\draw[blue][dashed](0,2.25)..controls(.7,2.25) and (.7,1.25) ..(0,1.25);
\draw[blue][dashed](0,-2.25)..controls(.7,-2.25) and (.7,-1.25) ..(0,-1.25);
\draw[blue][dashed](0,1.25)..controls (-1.7,1.25) and (-1.7,-1.25)..(0,-1.25);
\draw[red](0,2.25)..controls (3, 2.25) and (3,-2.25)..(0,-2.25);
\draw[red](0,2.25)..controls(-.7,2.25) and (-.7,1.25) ..(0,1.25);
\draw[red](0,-2.25)..controls(-.7,-2.25) and (-.7,-1.25) ..(0,-1.25);
\draw[red](0,1.25)..controls (1.7,1.25) and (1.7,-1.25)..(0,-1.25);
\end{tikzpicture}}
\end{figure}

\medskip  Label the intersection points  $a, b, c, e$ vertically down the diagram as in the figure. (Since we plan to discuss the differentials $d_i$ in the spectral sequence, we will not also use $d$ to label any intersection points.) There are no differentials that count for $\widehat{\mathit{HFL}}(\mathcal D)$ -- which is exactly the link Floer homology of the Hopf link with positive linking number -- and three differentials that count for $\widehat{\mathit{HFK}}(\mathcal D) = \widehat{\mathit{HFK}}(S^3, K) \otimes W$.  See the table of Figure \ref{Unknot Complexes Figure} for the Alexander gradings of these entries.


\begin{figure}
\caption{Alexander gradings and differentials for $\widehat{\mathit{CFL}}(\mathcal D)$ (on the left) and $\widehat{\mathit{CFK}}(\mathcal D)$ (on the right).}
\label{Unknot Complexes Figure}
\subfloat{
\centering
\begin{tikzpicture}
\path (-2,2)edge (1,2);
\path (-2,-1)edge (-2,2);
\path (-2,-1) edge (1,-1);
\path (1,-1) edge (1,2);
\path (-2, 1) edge (1,1);
\path (-2,0) edge (1,0);
\path (-1,2) edge (-1,-1);
\path (0,2) edge (0,-1);
\path (-2,2) edge (-1,1);
\node (b) at (-.5,.5){$b$};
\node (a) at (.5,.5){$a$};
\node (c) at (-.5,-.5){$c$};
\node (e) at ( .5,-.5){$e$};
\node at (-.5,1.5){$-\frac{1}{2}$};
\node at (.5,1.5){$\frac{1}{2}$};
\node at (-1.5,.5){$\frac{1}{2}$};
\node at (-1.5,-.5){$-\frac{1}{2}$};
\node at (-1.7,1.3){$A_2$};
\node at (-1.3,1.7){$A_1$};
\end{tikzpicture}}
\hspace{1 cm}
\subfloat{
\begin{tikzpicture}
\path (-2,2)edge (1,2);
\path (-2,-1)edge (-2,2);
\path (-2,-1) edge (1,-1);
\path (1,-1) edge (1,2);
\path (-2, 1) edge (1,1);
\path (-1,0) edge (1,0);
\path (-1,2) edge (-1,-1);
\path (0,2) edge (0,-1);
\node (b) at (-.5,.5){$b$};
\node (a) at (.5,.5){$a$};
\node (c) at (-.5,-.5){$c$};
\node (d) at ( .5,-.5){$e$};
\path[->,dashed](b)edge(c);
\path[->,dashed][bend left](a)edge(d);
\path[->,dashed][bend right](a)edge(d);
\node at (-.5,1.5){$-1$};
\node at (.5,1.5){$0$};
\node at (-1.5,1.5){$A_1$};
\end{tikzpicture}}
\end{figure}





\medskip

Now lift to a diagram $\widetilde{\mathcal D}$ for the same Hopf link, which has basepoints $w_0, z_0$ on the axis $U$ and $w_1^1, w_1^2, z_1^1, z_1^2$  on the lifted unknot $\widetilde{K}$.  These basepoints lie on $S^2 = \mathbb C \cup \{\infty\}$ as follows: $w_0$ and $z_0$ lie at $0$ and $\infty$ as previously, and $w_1^1, w_1^2, z_1^1, z_1^2$ lie at $i, -i, 1, -1$ respectively.  There are two curves $\alpha_1^1$ and $\alpha_1^2$ encircling the pairs $w_1^1, z_1^1$ and $w_1^2, z_1^2$, and two curves $\beta_1^1$ and $\beta_1^2$ encircling pairs $z_1^2, w_1^1$ and $z_1^1, w_1^2$.  The intersection points $a, b, c, e$ lift to eight points $a^1,a^2,b^1,b^2,c^1,c^2,e^1,e^2$ on the diagram in $\cup (\alpha_1^i \cap \beta_1^j)$.  (The numbering of each pair is arbitrarily determined by insisting that $a^i$ lie on $\alpha_1^i$, and so on.) The complex $\widehat{\mathit{CFL}}(\widetilde{\mathcal D})$ has eight generators, whose Alexander gradings are laid out in the left table of Figure \ref{Lifted Unknot Complexes Figure}; there are no differentials that count for the theory $\widetilde{\mathit{HFL}}(\widetilde{\mathcal D}) = \widehat{\mathit{HFL}}(S^3, \widetilde{K} \cup U) \otimes V_1$.  Allowing differentials that pass over the basepoint $z_0$, we obtain the complex of the right table Figure \ref{Lifted Unknot Complexes Figure} which computes $\widetilde{\mathit{HFK}}(\mathcal D) = \widehat{\mathit{HFK}}(S^3, \widetilde{K}) \otimes V_1 \otimes W$.  


\begin{figure}
\caption{Alexander gradings and differentials of $\widehat{\mathit{CFL}}(\widetilde{\mathcal D})$ (left) and $\widehat{\mathit{CFK}}(\widetilde{\mathcal D})$ (right).}
\label{Lifted Unknot Complexes Figure}
\subfloat{
\centering
\begin{tikzpicture}
\path(-3,2)edge(2,2);
\path(-3,2)edge(-3,-1);
\path(-3,-1)edge(2,-1);
\path(2,2)edge(2,-1);
\path(-2,2)edge(-2,-1);
\path(-1,2)edge(-1,-1);
\path(1,2)edge(1,-1);
\path(-3,1)edge(2,1);
\path(-3,0)edge(2,0);
\path(-3,2)edge(-2,1);
\node at (-2.7,1.3){$A_2$};
\node at (-2.3,1.7){$A_1$};
\node at (-1.5,1.5){$-\frac{3}{2}$};
\node at (0,1.5){$-\frac{1}{2}$};
\node at (1.5,1.5){$\frac{1}{2}$};
\node at (-2.5, .5){$\frac{1}{2}$};
\node at (-2.5,-.5){$-\frac{1}{2}$};
\node (1) at (-1.5,.5){$b^1b^2$};
\node (2) at (-1.5,-.5){$c^1c^2$};
\node (3) at (1.5,.5){$a^1a^2$};
\node (4) at (1.5,-.5){$e^1e^2$};
\node (5) at (-.5,.5){$b^1a^2$};
\node (6) at (.5,.5){$a^1b^2$};
\node (7) at (-.5,-.5){$c^1e^2$};
\node (8) at (.5,-.5){$e^1c^2$};
\end{tikzpicture}}
\hspace{1 cm}
\subfloat{
\centering
\begin{tikzpicture}
\path(-3,2)edge(2,2);
\path(-3,2)edge(-3,-1);
\path(-3,-1)edge(2,-1);
\path(2,2)edge(2,-1);
\path(-2,2)edge(-2,-1);
\path(-1,2)edge(-1,-1);
\path(1,2)edge(1,-1);
\path(-3,1)edge(2,1);
\path(-2,0)edge(2,0);
\node at (-2.5,1.5){$A_1$};
\node at (-1.5,1.5){$-2$};
\node at (0,1.5){$-1$};
\node at (1.5,1.5){$0$};
\node (1) at (-1.5,.5){$b^1b^2$};
\node (2) at (-1.5,-.5){$c^1c^2$};
\node (3) at (1.5,.5){$a^1a^2$};
\node (4) at (1.5,-.5){$e^1e^2$};
\node (5) at (-.5,.5){$b^1a^2$};
\node (6) at (.5,.5){$a^1b^2$};
\node (7) at (-.5,-.5){$c^1e^2$};
\node (8) at (.5,-.5){$e^1c^2$};
\path[->, dashed](1)edge(2);
\path[->,dashed][bend left](3)edge(4);
\path[->,dashed][bend right](3)edge(4);
\path[->,dashed](5)edge(7);
\path[->,dashed](5)edge(8);
\path[->,dashed](6)edge(7);
\path[->,dashed](6)edge(8);
\end{tikzpicture}}
\end{figure}

\medskip

First, consider the spectral sequence for link Floer homology of the double complex $(\widehat{\mathit{CFL}}(\widetilde{\mathcal D} \otimes \mathbb Z_2, \partial + (1 + \tau^{\#})\theta)$.  Since $\partial = 0$ on this complex, the only nontrivial differential occurs on the $E^1$ page and is exactly $d_1 = 1 + \tau^{\#}$.  Therefore the $E^2$ page of the spectral sequence is $\mathbb Z_2((\theta))\langle a^1a^2, b^1b^2, c^1c^2, e^1e^2 \rangle$, which is isomorphic to $\widehat{\mathit{HFL}}(S^3, K \cup U) \otimes \mathbb Z_2((\theta))$, as expected.

\medskip

Next, consider the spectral sequence of the double complex $(\widehat{\mathit{CFK}}(\widetilde{\mathcal D} \otimes \mathbb Z_2, \partial_{U} + (1 + \tau^{\#})\theta)$.  We have the complex of Figure \ref{Lifted Unknot Complexes Figure}; the $E^1$ page of the spectral sequence is equal to $\mathbb Z_2\langle [a^1a^2], [e^1e^2], [a^1b^2+b^1a^2], [c^1e^2] \rangle \otimes \mathbb Z_2((\theta))$.  Bearing in mind that $[c^1e^2] = [c^2e^1]$, we see that $\tau^*$ is the identity on each of these four elements, and therefore the $E^2$ page of the spectral sequence is the same as the $E^1$ page. Since the differential $d_2$ must raise the Maslov grading by two, the only possible nontrivial differential on the $E^2$ page is $d_2([c^1e^2]\theta^n)$.  Let us compute this differential.  On the chain level, we have $(1+ \tau^{\#})(c^1e^2) = (c^1e^2 + c^2e^1)$.  We observe that $c^1e^2 + c^2e^1 = \partial(a^1b^2)$, so 

\begin{align*}
d_2([c^1e^2]\theta^n) &= [(1+\tau^{\#})(a^1b^2)]\theta^{n+1} \\
				&=[a^1b^2 + a^2b^1]\theta^{n+2}
\end{align*}


Therefore the $E^3$ page of this spectral sequence is exactly $\mathbb Z_2((\theta))\langle[a^1a^2], [e^1e^2]\rangle$, which is isomorphic to $\left(\widehat{\mathit{HFK}}(S^3,K)\otimes W\right)\otimes \mathbb Z_2((\theta))$ as promised, and unchanged thereafter.

\FloatBarrier

\subsection{The case of the trefoil}

Let us now compute some of the spectral sequence for the trefoil as a doubly-periodic knot with quotient the unknot, using the Heegaard diagrams $\mathcal D$ and $\widetilde{\mathcal D}$ of Figure \ref{Trefoil Heegaard Diagrams Figure}.  Recall that $\mathcal D$ is a Heegaard diagram for a link $L$ in $S^3$ consisting of two unknots with linking number $\lambda = 3$ and $\widetilde{\mathcal D}$ is a Heegaard diagram for a link $\widetilde{L}$ in $S^3$ consisting of the left-handed trefoil and the unknotted axis also with linking number $\lambda=3$. We label the twelve intersection points of $\alpha_1$ and $\beta_1$ in $\mathcal D$ as shown in Figure \ref{Trefoil Quotient Generators}, and lift to twenty-four intersection points in $\widetilde{\mathcal D}$ in Figure \ref{Trefoil Generators}.  The Alexander gradings and differentials of $\widehat{\mathit{CFL}}(\mathcal D)$ and $\widehat{\mathit{CFK}}(\mathcal D)$ are laid out in the tables of Figure \ref{Trefoil Quotient Complexes}.

\begin{figure}
\centering
\caption{Intersection points in $\mathcal D$.}
\label{Trefoil Quotient Generators}
\begin{tikzpicture}
\tikzstyle{every node}=[font=\small]
\node(2) at (-4,1){$z_1$};
\node(3) at (-1,0){$w_1$};
\tikzstyle{every node}=[font=\tiny]
\node(4) at (-4.5,1.8){$a$};
\node(5) at (-3.8,1.9){$b$};
\node(6) at (-.5,-.8){$h$};
\node(7) at (-1.15,-1){$i$};
\node(8) at (-1.5,-1){$j$};
\node(9) at (-1.5,1.4){$g$};
\node(10) at (-2.1,1.65){$f$};
\node(11) at (-2.1,-.9){$k$};
\node(12) at (-2.65,-.75){$\ell$};
\node(13) at (-2.6,1.8){$e$};
\node(14) at (-3.1,1.9){$c$};
\node(15) at (-3.2,-.55){$m$};
\def \alphacurve {[rotate=-18.5](-2.53,-.32) ellipse (65pt and 28pt)};
\draw [blue][dashed][line width = 2 pt] \alphacurve;
\clip \alphacurve;
\tikzset{every path/.style={line width = 2 pt}};
\draw[red](-4.3,1) ..controls (-4.5,6.5) and (3.4,6.5)..(3.3,.3);
\draw[red](3.3,.3)..controls (3.4, -4.9) and (-3.4,-5)..(-3.3,.7);
\draw[red](-3.3,.7)..controls (-3.4, 5) and (2.4,5)..(2.3, .5);
\draw[red](2.3, .5)..controls (2.4, -3.5) and (-2.4, -3.5)..(-2.3, .4);
\draw[red](-2.3,.4)..controls (-2.4, 3.5) and (1.4, 3.4)..(1.3,.7);
\draw[red](1.3,.7)..controls (1.4,-2.15) and (-1.4, -2.15)..(-1.3,0);
\draw[red](-1.3,0)..controls (-1.4, .5) and (-.7,.5)..(-.7,0);
\draw[red](-.7,0)..controls (-.8,-1.3) and (.8, -1.2)..(.7,.5);
\draw[red](.7,.5)..controls (.8,2.5) and (-1.8, 2.5)..(-1.7,.4); 
\draw[red](-1.7,.4)..controls(-1.8,-2.65) and (1.8, -2.65) ..(1.7,.6);
\draw[red](1.7,.6)..controls(1.8,4) and(-2.8,4) ..(-2.7,.5);
\draw[red](-2.7,.5)..controls(-2.8,-4.1) and (2.8,-4.1) ..(2.7,.5);
\draw[red](2.7,.5)..controls (2.85,5.55) and (-3.8,5.75)..(-3.7,.9);
\draw[red](-3.7,.9)..controls (-3.8,.6) and (-4.4,.6) .. (-4.3,1);
\end{tikzpicture}
\end{figure}


\begin{figure}
\caption{Intersection points of $\alpha$ and $\beta$ curves in $\widetilde{\mathcal D}$.}
\label{Trefoil Generators}
\centering
\subfloat{
\begin{tikzpicture}
\tikzstyle{every node}=[font=\small]
\node(1) at (0,.7){$z_0$};
\node(2) at (-4,1){$z_1^1$};
\node(3) at (-1,0){$w_1^1$};
\tikzstyle{every node}=[font=\tiny]
\node(4) at (-4.4,1.7){$a^1$};
\node(5) at (-3.7,1.8){$b^1$};
\node(6) at (-.45,-.7){$h^1$};
\node(7) at (-1.05,-.85){$i^1$};
\node(8) at (-1.5,-.85){$j^1$};
\node(9) at (-1.4,1.3){$g^1$};
\node(10) at (-2,1.55){$f^1$};
\node(11) at (-2.1,-.75){$k^1$};
\node(12) at (-2.55,-.65){$\ell^1$};
\node(13) at (-2.5,1.7){$e^1$};
\node(14) at (-3,1.8){$c^1$};
\node(15) at (-3.1,-.45){$m^1$};
\def \betacurve1{[rotate=-18.5] (-2.53,-.32) ellipse (61pt and 24pt)};
\clip \betacurve1;
\tikzset{every path/.style={line width = 2 pt}}
\draw [rotate=-18.5][blue][dashed](-2.53,-.32) ellipse (60pt and 23pt);
\draw [rotate=-18.5][cyan][dashed] (2.21,1.27) ellipse (60pt and 23pt);
\draw[red](-4.3,1)..controls (-4.4,6.65) and (3.4,6.4)..(3.3,.5);
\draw[red](3.3,.5)..controls (3.3,-4.2) and (-2.3,-3.9)..(-2.3,.2);
\draw[red](-2.3,.2)..controls (-2.4,3.6) and (1.4,3.6)..(1.3,1);
\draw[red](1.3,1)..controls (1.4,.5) and (.7,.5) ..(.7,1);
\draw[red](.7,1)..controls (.8,2.8) and (-1.7,2.5)..(-1.7,.5);
\draw[red](-1.7,.5)..controls (-1.8,-3.2) and (2.7, -3.2)..(2.7,.4);
\draw[red](2.7,.4)..controls (2.7, 5.6) and (-3.85, 5.8) ..(-3.7,.9);
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\draw[magenta](3.7,.2)..controls (3.7,.5) and (4.3,.5)..(4.3,0);
\end{tikzpicture}}
\subfloat{
\begin{tikzpicture}
\tikzstyle{every node}=[font=\small]
\node(4) at (1,1){$w_1^2$};
\node(5) at (4,0){$z_1^2$};
\tikzstyle{every node}=[font=\tiny]
\node(6) at (.6,1.7){$h^2$};
\node(7) at (1.2,1.8){$i^2$};
\node(8) at (1.7,1.8){$j^2$};
\node(9) at (2.3,1.7){$k^2$};
\node(10) at (2.8,1.55){$\ell^2$};
\node(11) at (3.5,1.3){$m^2$};
\node(12) at (4.5,-.65){$a^2$};
\node(13) at (3.8,-.8){$b^2$};
\node(14) at (3.15,-.8){$c^2$};
\node(15) at (2.65,-.7){$e^2$};
\node(12) at (2.1,-.6){$f^2$};
\node(15) at (1.7,-.4){$g^2$};
\def \betacurve2{[rotate=-18.5] (2.21,1.27) ellipse (61pt and 24pt)};
\clip \betacurve2;
\tikzset{every path/.style={line width = 2 pt}}
\draw [rotate=-18.5][blue][dashed](-2.53,-.32) ellipse (60pt and 23pt);
\draw [rotate=-18.5][cyan][dashed] (2.21,1.27) ellipse (60pt and 23pt);
\draw[red](-4.3,1)..controls (-4.4,6.65) and (3.4,6.4)..(3.3,.5);
\draw[red](3.3,.5)..controls (3.3,-4.2) and (-2.3,-3.9)..(-2.3,.2);
\draw[red](-2.3,.2)..controls (-2.4,3.6) and (1.4,3.6)..(1.3,1);
\draw[red](1.3,1)..controls (1.4,.5) and (.7,.5) ..(.7,1);
\draw[red](.7,1)..controls (.8,2.8) and (-1.7,2.5)..(-1.7,.5);
\draw[red](-1.7,.5)..controls (-1.8,-3.2) and (2.7, -3.2)..(2.7,.4);
\draw[red](2.7,.4)..controls (2.7, 5.6) and (-3.85, 5.8) ..(-3.7,.9);
\draw[red](-3.7,.9)..controls (-3.7,.5) and (-4.3,.5) ..(-4.3,1);
\draw[magenta](4.3,0)..controls (4.4,-5.65) and (-3.4,-5.4)..(-3.3,.65);
\draw[magenta](-3.3,.65)..controls (-3.3,4.9) and (2.45,5.25) ..(2.3,.3);
\draw[magenta](2.3,.3)..controls (2.2,-2.45) and (-1.4,-2.55)..(-1.3,0);
\draw[magenta](-1.3,0)..controls (-1.3,.5) and (-.7,.5)..(-.7,0);
\draw[magenta](-.7,0)..controls (-.8,-1.8) and (1.7,-1.5)..(1.7,.6);
\draw[magenta](1.7,.6)..controls (1.8,4.2) and (-2.7,4.2) ..(-2.7,.6);
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\draw[magenta](3.7,.2)..controls (3.7,.5) and (4.3,.5)..(4.3,0);
\end{tikzpicture}}
\end{figure}






\begin{figure}
\caption{Alexander gradings and differentials of $\widehat{\mathit{CFL}}(\mathcal D)$ (left) and $\widehat{\mathit{CFK}}(\mathcal D)$ (right).}
\label{Trefoil Quotient Complexes}
\subfloat{\centering
\begin{tikzpicture}
\draw(-3,3)edge(4,3);
\draw(-3,3)edge(-3,-2);
\draw(-3,-2)edge(4,-2);
\draw(4,-2)edge(4,3);
\draw(-3,2)edge(4,2);
\draw(-3,1)edge(4,1);
\draw(-3,0)edge(4,0);
\draw(-3,-1)edge(4,-1);
\draw(-2,3) edge (-2,-2);
\draw(-1,3)edge(-1,-2);
\draw(1,3)edge (1,-2);
\draw(3,3)edge(3,-2);
\draw(-3,3)edge(-2,2);
\node at (-2.7,2.3){$A_2$};
\node at (-2.3,2.7){$A_1$};
\node at (-1.5,2.5){$-\frac{3}{2}$};
\node at (0, 2.5){$-\frac{1}{2}$};
\node at (2, 2.5){$\frac{1}{2}$};
\node at (3.5,2.5){$\frac{3}{2}$};
\node at (-2.5,1.5){$\frac{3}{2}$};
\node at (-2.5,.5){$\frac{1}{2}$};
\node at (-2.5,-.5){$-\frac{1}{2}$};
\node at (-2.5,-1.5){$-\frac{3}{2}$};
\node(1) at (3.5,-.5){$m$};
\node(2) at (3.5,-1.5){$a$};
\node(3) at (2,.5){$k$};
\node(4) at (1.5,-.5){$c$};
\node(5) at (2.5,-.5){$\ell$};
\node(6) at (2,-1.5){$b$};
\node(7) at (0,1.5){$i$};
\node(8) at (-.5,.5){$f$};
\node(9) at (.5,.5){$j$};
\node(10) at (0,-.5){$e$};
\node(11) at (-1.5,1.5){$h$};
\node(12) at (-1.5,.5){$g$};  
\end{tikzpicture}}
\hspace{1 cm}
\subfloat{\centering
\begin{tikzpicture}
\draw(-3,3)edge(4,3);
\draw(-3,3)edge(-3,-2);
\draw(-3,-2)edge(4,-2);
\draw(4,-2)edge(4,3);
\draw(-3,2)edge(4,2);
\draw(-2,1)edge(4,1);
\draw(-2,0)edge(4,0);
\draw(-2,-1)edge(4,-1);
\draw(-2,3) edge (-2,-2);
\draw(-1,3)edge(-1,-2);
\draw(1,3)edge (1,-2);
\draw(3,3)edge(3,-2);
\node at (-2.5,2.5){$A_1$};
\node at (-1.5,2.5){$-3$};
\node at (0, 2.5){$-2$};
\node at (2, 2.5){$-1$};
\node at (3.5,2.5){$0$};
\node(1) at (3.5,-.5){$m$};
\node(2) at (3.5,-1.5){$a$};
\node(3) at (2,.5){$k$};
\node(4) at (1.5,-.5){$c$};
\node(5) at (2.5,-.5){$\ell$};
\node(6) at (2,-1.5){$b$};
\node(7) at (0,1.5){$i$};
\node(8) at (-.5,.5){$f$};
\node(9) at (.5,.5){$j$};
\node(10) at (0,-.5){$e$};
\node(11) at (-1.5,1.5){$h$};
\node(12) at (-1.5,.5){$g$}; 
\path[->,dashed][bend left](1)edge(2);
\path[->,dashed][bend right](1)edge(2);
\path[->,dashed](3)edge(4);
\path[->,dashed](3)edge(5);
\path[->,dashed](4)edge(6);
\path[->,dashed](5)edge(6);
\path[->,dashed](7)edge(8);
\path[->,dashed](7)edge(9);
\path[->,dashed](8)edge(10);
\path[->,dashed](9)edge(10);
\path[->,dashed](11)edge(12);
\end{tikzpicture}}
\end{figure}

There are no differentials on $\widehat{\mathit{CFL}}(\mathcal D)$, so $\widetilde{\mathit{HFL}}(\mathcal D) = \widehat{\mathit{HFL}}(S^3,L)$ has the twelve generators and gradings of Figure \ref{Trefoil Quotient Generators}.  From the remainder of that diagram, we observe that the group $\widetilde{\mathit{HFK}}(\mathcal D) = \widehat{\mathit{HFK}}(S^3,U) \otimes W$ is $\mathbb Z_2 \langle [a],[m] \rangle$.

\medskip

Now consider the seventy-two generators of $\widehat{\mathit{CFL}}(\widetilde{\mathcal D})$, whose Alexander $A_1$ and $A_2$ gradings are laid out in Figure \ref{CFL(trefoil) Figure}.  It so happens that $\widetilde{\mathcal D}$ is a nice diagram in the sense of Sarkar and Wang \cite{MR2630063}, although the equivariant diagrams for periodic knots introduced in Section \ref{Periodic Knots Section} are not in general, so we may compute $\widetilde{\mathit{HFL}}(\widetilde{\mathcal D})$ with relative ease.  The chain complexes in each Alexander $A_1$ grading are shown in Figures \ref{-7/2 Figure}, \ref{-5/2 Figure}, \ref{-3/2 Figure}, \ref{-1/2 Figure}, \ref{1/2 Figure}, \ref{3/2 Figure} and \ref{5/2 Figure} at the close of this section.  These convention for these figures is as follows: the generators shown are those in the $A_1$ grading of $\widehat{\mathit{CFL}}(\widetilde{\mathcal D})$, which are also the generators of the $A_1 - \frac{3}{2}$ grading of $\widehat{\mathit{CFK}}(\widetilde{\mathcal D})$.  Solid arrows denote differentials that count for the differential $\partial$ and thus exist in both complexes, whereas dashed arrows denote differentials corresponding to disks with nontrivial intersection with the divisor $V_{z_0} = \{z_0\}\times \text{Sym}^{2n_1}(S^2)$.  Therefore dashed differentials only count for the knot Floer complex $(\widehat{\mathit{CFK}}(\widetilde{\mathcal D}), \partial_U)$.



\begin{figure}
\caption{Alexander $A_1$ and $A_2$ gradings of the generators of $\widehat{\mathit{CFL}}(\widetilde{D})$.  These generate the $E^0$ page of the link Floer homology spectral sequence.}
\label{CFL(trefoil) Figure}
\centering
\begin{tikzpicture}[scale=.9]
\draw(-8,20)edge(7,20);
\draw(-8,20)edge(-8,9);
\draw(7,20)edge(7,9);
\draw(-8,9)edge(7,9);
\draw(-8,19)edge(7,19);
\draw(-8,20)edge(-7,19);
\node at (-7.7,19.3){$A_2$};
\node at (-7.3,19.7){$A_1$};
\draw(-7,20)edge(-7,9);
\node at (-6,19.5){$-\frac{7}{2}$};
\node at (-5.5,18){$ $};
\node at (-6,18.25)[font = \small]{$h^1h^2$};
\node at (-6,15.75)[font = \small]{$g^1g^2$};
\draw(-5,20)edge(-5,9);
\node at (-4,19.5){$-\frac{5}{2}$};
\node at (-4.5,18.25)[font = \small]{$h^1i^2$};
\node at (-3.5,18.25)[font=\small]{$i^1h^2$};
\draw(-8,17.5)edge(7,17.5);
\node at (-4.5, 16.25)[font = \small]{$g^1j^2$};
\node at (-3.5, 16.25)[font=\small]{$j^1g^2$};
\node at (-4.5, 15.75)[font=\small] {$g^1f^2$};
\node at (-3.5, 15.75)[font=\small]{$f^1g^2$};
\node at (-4.5, 15.25)[font=\small]{$h^1e^2$};
\node at (-3.5, 15.25)[font=\small]{$e^1h^2$};
\draw (-3,20)edge(-3,9);
\node at (-7.5,15.75){$\frac{1}{2}$};
\node at (-7.5,18.25){$\frac{3}{2}$};
\node at (-2.5,17)[font=\small]{$f^1f^2$};
\node at (-1.5,17)[font = \small]{$j^1j^2$};
\node at (-2.5,16.5)[font = \small]{$e^1i^2$};
\node at (-1.5,16.5)[font = \small]{$i^1e^2$};
\node at (-2.5,16)[font = \small]{$f^1j^2$};
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\node at (-2.5,15.5)[font = \small]{$g^1k^2$};
\node at (-1.5,15.5)[font = \small]{$k^1g^2$};
\node at (-2.5,15)[font = \small]{$h^1c^2$};
\node at (-1.5,15)[font=\small]{$h^2c^1$};
\node at (-2.5,14.5)[font = \small]{$h^1 \ell^2$};
\node at (-1.5,14.5)[font = \small]{$\ell^1 h^2$};
\node at (-2,19.5)[font = \small]{$-\frac{3}{2}$};
\node at (-2,18.25)[font=\small]{$i^1i^2$};
\draw(-1,20)edge(-1,9);
\draw(-8,14)edge(7,14);
\node at (-.5 ,13.25)[font = \small]{$e^1c^2$};
\node at (.5,13.25)[font=\small]{$c^1e^2$}; 
\node at (-.5,12.75)[font = \small]{$e^1\ell^2$};
\node at (.5,12.75)[font = \small]{$\ell^1e^2$};
\node at (-.5,12.25)[font = \small]{$b^1f^2$};
\node at (.5, 12.25)[font=\small]{$f^1b^2$};
\node at (-.5,11.75)[font = \small]{$b^1j^2$};
\node at (.5,11.75)[font = \small]{$j^1b^2$};
\node at (-.5,11.25)[font = \small]{$a^1g^2$};
\node at (.5,11.25)[font = \small]{$g^1a^2$};
\draw(1,20)edge(1,9);
\node at (0,19.5){$-\frac{1}{2}$};
\draw(-8,10.5)edge(7,10.5);
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\node at (-2,12.5)[font = \small]{$e^1e^2$};
\node at (-2.5,12)[font=\small]{$b^1g^2$};
\node at (-1.5,12)[font=\small]{$g^1b^2$};
\node at (-.5,16.75)[font=\small]{$f^1k^2$};
\node at (.5,16.75)[font=\small]{$k^1f^2$};
\node at (-.5,16.25)[font=\small]{$j^1k^2$};
\node at (.5,16.25)[font=\small]{$k^1j^2$};
\node at (-.5,15.75)[font = \small]{$i^1\ell^2$};
\node at (.5,15.75)[font=\small]{$\ell^1i^2$};
\node at (-.5,15.25)[font=\small]{$c^1i^2$};
\node at (.5,15.25)[font=\small]{$i^1c^2$};
\node at (-.5,14.75)[font = \small]{$h^1m^2$};
\node at (.5,14.75)[font = \small]{$m^1h^2$};
\draw(3,20)edge(3,9);
\node at (2,19.5) {$\frac{1}{2}$};
\draw(5,20)edge(5,9);
\node at (4,19.5) {$\frac{3}{2}$};
\node at (6,19.5){$\frac{5}{2}$};
\node at (6,12.25)[font = \small]{$m^1m^2$};
\node at (2,16)[font=\small]{$k^1k^2$};
\node at (1.5,15.5)[font = \small]{$i^1m^2$};
\node at (2.5,15.5)[font = \small]{$m^1i^2$};
\node at (3.5,12.75)[font = \small]{$c^1m^2$};
\node at (4.5,12.75)[font = \small]{$m^1c^2$};
\node at (3.5,12.25)[font = \small]{$\ell^1 m^2$};
\node at (4.5,12.25)[font = \small]{$m^1 \ell^2$};
\node at (3.5,11.75)[font=\small]{$a^1k^2$};
\node at (4.5,11.75)[font=\small]{$k^1a^2$};
\node at (-7.5,9.75) {$-\frac{3}{2}$};
\node at (6,9.75)[font = \small]{$a^1a^2$};
\node at (3.5,9.75)[font = \small]{$a^1b^2$};
\node at (4.5,9.75)[font = \small]{$b^1a^2$};
\node at (2,9.75)[font=\small]{$b^1b^2$};
\node at (1.5, 12.5)[font = \small]{$b^1k^2$};
\node at (2.5, 12.5)[font = \small]{$k^1b^2$};
\node at (1.5,12)[font = \small]{$a^1f^2$};
\node at (2.5,12)[font = \small]{$f^1a^2$};
\node at (1.5,13.5)[font = \small]{$c^1c^2$};
\node at (2.5,13.5)[font = \small]{$\ell^1\ell^2$};
\node at (1.5,13)[font = \small]{$c^1 \ell^2$};
\node at (2.5,13)[font = \small]{$\ell^1 c^2$};
\node at (1.5,11.5)[font = \small]{$e^1 m^2$};
\node at (2.5,11.5)[font = \small]{$m^1e^2$};
\node at (1.5,11)[font = \small]{$a^1j^2$};
\node at (2.5,11)[font = \small]{$j^1a^2$};
\end{tikzpicture}
\end{figure}


\medskip
The link Floer homology spectral sequence associated to $\widetilde{\mathcal D}$ arises from the double complex $(\widehat{\mathit{CFL}}(\widetilde{\mathcal D}), \partial + (1 + \tau^{\#})\theta)$.
Computing homology of $\widehat{\mathit{CFL}}(\widetilde{\mathcal D})$ with respect to the differential $\partial$, we obtain a set of generators for the $E^1$ page of the spectral sequence, which is $\widetilde{\mathit{HFL}}(\widetilde{\mathcal D}) \otimes \mathbb Z_2((\theta)) = (\widehat{\mathit{HFL}}(S^3, 3_1 \cup U) \otimes V_1) \otimes \mathbb Z_2((\theta))$. These generators and their gradings may be found in Figure \ref{HFL(trefoil) figure}.  Whenever an element of $\widetilde{\mathit{HFK}}(\widetilde{\mathcal D})$ is invariant under the involution $\tau^*$ but has no representative which is invariant under the chain map $\tau^{\#}$, we have included two representatives of that element to make the $\tau^*$ invariance clear.  (For example, observe that $[h^1c^2] = [c^1h^2]$ is invariant under $\tau^*$.)


\begin{figure}
\caption{The homology $\widetilde{\mathit{HFL}}(\widetilde{\mathcal D})$.  These elements generate the $E^1$ page of the link Floer spectral sequence as a $\mathbb Z_2((\theta))$ module.}
\label{HFL(trefoil) figure}
\centering
\begin{tikzpicture}[scale=.9]
\draw(-7.75,20)edge(8.25,20);
\draw(-7.75,20)edge(-7.75,9);
\draw(8.25,20)edge(8.25,9);
\draw(-7.75,9)edge(8.25,9);
\draw(-7.75,19)edge(8.25,19);
\draw(-7.75,20)edge(-6.75,19);
\draw(-6.75,20)edge(-6.75,9);
\draw(-5.25,20)edge(-5.25,9);
\draw(-7.75,17.5)edge(8.25,17.5);
\draw (-3,20)edge(-3,9);
\draw(-.5,20)edge(-.5,9);
\draw(-7.75,14)edge(8.25,14);
\draw(1.75,20)edge(1.75,9);
\draw(-7.75,10.5)edge(8.25,10.5);
\draw(4.25,20)edge(4.25,9);
\draw(6.75,20)edge(6.75,9);
\node at (-7.5,19.3){$A_2$};
\node at (-7.1,19.7){$A_1$};
\node at (-6,19.5){$-\frac{7}{2}$};
\node at (-6,18.25)[font = \small]{$[h^1h^2]$};
\node at (-6,15.75)[font = \small]{$[g^1g^2]$};
\node at (-4.2,19.5){$-\frac{5}{2}$};
\node at (-4.65,18.25)[font = \small]{$[h^1i^2]$};
\node at (-3.65,18.25)[font=\small]{$[i^1h^2]$};
\node at (-4.15, 16.25)[font = \small]{$[g^1j^2+f^1g^2]$};
\node at (-4.15, 15.75)[font=\small]{$[j^1g^2+g^1f^2]$};
\node at (-7.25,15.75){$\frac{1}{2}$};
\node at (-7.25,18.25){$\frac{3}{2}$};
\node at (-1.75,16.5)[font=\small]{$[f^1f^2 + k^1g^2 +$};
\node at (-1.75,16)[font = \small]{$ g^1k^2 + j^1j^2]$};
\node at (-1.75,15.25)[font = \small]{$[h^1c^2] = [c^1h^2]$};
\node at (-1.75,19.5)[font = \small]{$-\frac{3}{2}$};
\node at (-1.75,18.25)[font=\small]{$[i^1i^2]$};
\node at (.1,12.25)[font = \small]{$[a^1g^2]$};
\node at (1.2,12.25)[font = \small]{$[g^1a^2]$};
\node at (.6,19.5){$-\frac{1}{2}$};
\node at (-7.25,12.25){$-\frac{1}{2}$};
\node at (-1.75,12.25)[font=\small]{$[b^1g^2+g^1b^2]$};
\node at (1.2,15.75)[font=\small]{$[c^1i^2]$};
\node at (.1,15.75)[font=\small]{$[i^1c^2]$};
\node at (3,19.5) {$\frac{1}{2}$};
\node at (5.4,19.5) {$\frac{3}{2}$};
\node at (7.5,19.5){$\frac{5}{2}$};
\node at (7.5,12.25)[font = \small]{$[m^1m^2]$};
\node at (3,15.75)[font = \small]{$[i^1m^2] = [m^1i^2]$};
\node at (5.5,12.5)[font = \small]{$[c^1m^2+m^1\ell^2]$};
\node at (5.5,12)[font = \small]{$[m^1c^2+\ell^1 m^2]$};
\node at (-7.25,9.75) {$-\frac{3}{2}$};
\node at (7.5,9.75)[font = \small]{$[a^1a^2]$};
\node at (4.9,9.75)[font = \small]{$[a^1b^2]$};
\node at (6,9.75)[font = \small]{$[b^1a^2]$};
\node at (3,9.75)[font=\small]{$[b^1b^2]$};
\node at (3,11.5)[font = \small]{$[a^1f^2]=[f^1a^2]$};
\node at (3,13)[font = \small]{$[c^1c^2+e^1 m^2 +$};
\node at (3,12.5)[font = \small]{$m^1e^2 + \ell^1 \ell^2]$};
\end{tikzpicture}
\end{figure}

\medskip

The differential $d_1$ on the $E_1$ page of the link Floer spectral sequence for $\widetilde{\mathcal D}$ is $(1 + \tau^*)\theta$; in particular, computing the homology of $d_1$ has the effect of killing all elements of $\widetilde{\mathit{HFL}}(\mathcal D)$ not invariant under $\tau^*$.  Ergo we see that the $E^2$ page of this spectral sequence is generated as a $\mathbb Z_2((\theta))$ module by the elements of Figure \ref{HFL(E2) Figure}.  Notice that the ranks of the $E^2$ page in each Alexander grading $A_1 = 2k + \frac{1}{2}$ correspond precisely to the ranks of each Alexander grading $A_1 = k + \frac{1}{2}$ of Figure \ref{Trefoil Quotient Complexes}, which is the link Floer homology $\widetilde{\mathit{HFL}}(\mathcal D)$.  Therefore the link Floer homology spectral sequence converges on the $E^2$ page.

\begin{figure}
\caption{Generators for the $E^2 = E^{\infty}$ page of the link Floer spectral sequence for $\widetilde{\mathcal D}$ as a $\mathbb Z_2((\theta))$-module.}
\label{HFL(E2) Figure}
\centering
\begin{tikzpicture}[scale=.9]
\draw(-7.75,20)edge(8.25,20);
\draw(-7.75,20)edge(-7.75,9);
\draw(8.25,20)edge(8.25,9);
\draw(-7.75,9)edge(8.25,9);
\draw(-7.75,19)edge(8.25,19);
\draw(-7.75,20)edge(-6.75,19);
\draw(-6.75,20)edge(-6.75,9);
\draw(-5.25,20)edge(-5.25,9);
\draw(-7.75,17.5)edge(8.25,17.5);
\draw (-3,20)edge(-3,9);
\draw(-.5,20)edge(-.5,9);
\draw(-7.75,14)edge(8.25,14);
\draw(1.75,20)edge(1.75,9);
\draw(-7.75,10.5)edge(8.25,10.5);
\draw(4.25,20)edge(4.25,9);
\draw(6.75,20)edge(6.75,9);
\node at (-7.5,19.3){$A_2$};
\node at (-7.1,19.7){$A_1$};
\node at (-6,19.5){$-\frac{7}{2}$};
\node at (-6,18.25)[font = \small]{$[h^1h^2]$};
\node at (-6,15.75)[font = \small]{$[g^1g^2]$};
\node at (-4.2,19.5){$-\frac{5}{2}$};
\node at (-7.25,15.75){$\frac{1}{2}$};
\node at (-7.25,18.25){$\frac{3}{2}$};
\node at (-1.75,16.5)[font=\small]{$[f^1f^2 + k^1g^2 +$};
\node at (-1.75,16)[font = \small]{$ g^1k^2 + j^1j^2]$};
\node at (-1.75,15.25)[font = \small]{$[h^1c^2] = [c^1h^2]$};
\node at (-1.75,19.5)[font = \small]{$-\frac{3}{2}$};
\node at (-1.75,18.25)[font=\small]{$[i^1i^2]$};
\node at (.6,19.5){$-\frac{1}{2}$};
\node at (-7.25,12.25){$-\frac{1}{2}$};
\node at (-1.75,12.25)[font=\small]{$[b^1g^2+g^1b^2]$};
\node at (3,19.5) {$\frac{1}{2}$};
\node at (5.4,19.5) {$\frac{3}{2}$};
\node at (7.5,19.5){$\frac{5}{2}$};
\node at (7.5,12.25)[font = \small]{$[m^1m^2]$};
\node at (3,15.75)[font = \small]{$[i^1m^2] = [m^1i^2]$};
\node at (-7.25,9.75) {$-\frac{3}{2}$};
\node at (3,9.75)[font=\small]{$[b^1b^2]$};
\node at (3,11.5)[font = \small]{$[a^1f^2]=[f^1a^2]$};
\node at (3,13)[font = \small]{$[c^1c^2+e^1 m^2 +$};
\node at (3,12.5)[font = \small]{$m^1e^2 + \ell^1 \ell^2]$};
\node at (7.5,9.75)[font = \small]{$[a^1a^2]$};
\end{tikzpicture}
\end{figure}


\medskip

We now turn our attention to the knot Floer homology spectral sequence for $\widetilde{\mathcal D}$, which arises from the double complex $(\widehat{\mathit{CFK}}(\mathcal D) \otimes \mathbb Z_2((\theta)), \partial_U + (1 + \tau^{\#})\theta))$. The Alexander $A_1$ gradings of the seventy-two generators in $\widehat{\mathit{CFK}}(\widetilde{\mathcal D})$ are laid out in Figure \ref{CFK(trefoil) Figure}.  Notice that these gradings are exactly the $A_1$ gradings of $\widehat{\mathit{CFL}}(\widetilde{\mathcal D})$ shifted downward by $\frac{\ell k(3_1, U)}{2} = \frac{3}{2}$.  These elements generate the $E^0$ page of the link Floer spectral sequence as a $\mathbb Z_2((\theta))$ module.


\begin{figure}
\caption{Alexander $A_1$ gradings of elements of $\widehat{\mathit{CFK}}(\widetilde{\mathcal D})$. These elements generate the $E^0$ page of the link Floer spectral sequence as a $\mathbb Z_2((\theta))$ module.}
\centering
\begin{tikzpicture}[scale=.9]
\draw(-8,20)edge(7,20);
\draw(-8,20)edge(-8,9);
\draw(7,20)edge(7,9);
\draw(-8,9)edge(7,9);
\draw(-8,19)edge(7,19);
\draw(-7,20)edge(-7,9);
\draw(-5,20)edge(-5,9);
\draw(-7,17.5)edge(7,17.5);
\draw (-3,20)edge(-3,9);
\draw(-1,20)edge(-1,9);
\draw(-7,14)edge(7,14);
\draw(1,20)edge(1,9);
\draw(-7,10.5)edge(7,10.5);
\draw(3,20)edge(3,9);
\draw(5,20)edge(5,9);
\node at (-7.5,19.5){$A_1$};
\node at (-6,19.5){$-5$};
\node at (-6,18.25)[font = \small]{$h^1h^2$};
\node at (-6,15.75)[font = \small]{$g^1g^2$};
\node at (-4,19.5){$-4$};
\node at (-4.5,18.25)[font = \small]{$h^1i^2$};
\node at (-3.5,18.25)[font=\small]{$i^1h^2$};
\node at (-4.5, 16.25)[font = \small]{$g^1j^2$};
\node at (-3.5, 16.25)[font=\small]{$j^1g^2$};
\node at (-4.5, 15.75)[font=\small] {$g^1f^2$};
\node at (-3.5, 15.75)[font=\small]{$f^1g^2$};
\node at (-4.5, 15.25)[font=\small]{$h^1e^2$};
\node at (-3.5, 15.25)[font=\small]{$e^1h^2$};
\node at (-2.5,17)[font=\small]{$f^1f^2$};
\node at (-1.5,17)[font = \small]{$j^1j^2$};
\node at (-2.5,16.5)[font = \small]{$e^1i^2$};
\node at (-1.5,16.5)[font = \small]{$i^1e^2$};
\node at (-2.5,16)[font = \small]{$f^1j^2$};
\node at (-1.5,16)[font = \small]{$j^1f^2$};
\node at (-2.5,15.5)[font = \small]{$g^1k^2$};
\node at (-1.5,15.5)[font = \small]{$k^1g^2$};
\node at (-2.5,15)[font = \small]{$h^1c^2$};
\node at (-1.5,15)[font=\small]{$h^2c^1$};
\node at (-2.5,14.5)[font = \small]{$h^1 \ell^2$};
\node at (-1.5,14.5)[font = \small]{$\ell^1 h^2$};
\node at (-2,19.5)[font = \small]{$-3$};
\node at (-2,18.25)[font=\small]{$i^1i^2$};
\node at (-.5 ,13.25)[font = \small]{$e^1c^2$};
\node at (.5,13.25)[font=\small]{$c^1e^2$}; 
\node at (-.5,12.75)[font = \small]{$e^1\ell^2$};
\node at (.5,12.75)[font = \small]{$\ell^1e^2$};
\node at (-.5,12.25)[font = \small]{$b^1f^2$};
\node at (.5, 12.25)[font=\small]{$f^1b^2$};
\node at (-.5,11.75)[font = \small]{$b^1j^2$};
\node at (.5,11.75)[font = \small]{$j^1b^2$};
\node at (-.5,11.25)[font = \small]{$a^1g^2$};
\node at (.5,11.25)[font = \small]{$g^1a^2$};
\node at (0,19.5){$-2$};
\node at (-2,12.5)[font = \small]{$e^1e^2$};
\node at (-2.5,12)[font=\small]{$b^1g^2$};
\node at (-1.5,12)[font=\small]{$g^1b^2$};
\node at (-.5,16.75)[font=\small]{$f^1k^2$};
\node at (.5,16.75)[font=\small]{$k^1f^2$};
\node at (-.5,16.25)[font=\small]{$j^1k^2$};
\node at (.5,16.25)[font=\small]{$k^1j^2$};
\node at (-.5,15.75)[font = \small]{$i^1\ell^2$};
\node at (.5,15.75)[font=\small]{$\ell^1i^2$};
\node at (-.5,15.25)[font=\small]{$c^1i^2$};
\node at (.5,15.25)[font=\small]{$i^1c^2$};
\node at (-.5,14.75)[font = \small]{$h^1m^2$};
\node at (.5,14.75)[font = \small]{$m^1h^2$};
\node at (2,19.5) {$-1$};
\node at (4,19.5) {$0$};
\node at (6,19.5){$1$};
\node at (6,12.25)[font = \small]{$m^1m^2$};
\node at (2,16)[font=\small]{$k^1k^2$};
\node at (1.5,15.5)[font = \small]{$i^1m^2$};
\node at (2.5,15.5)[font = \small]{$m^1i^2$};
\node at (3.5,12.75)[font = \small]{$c^1m^2$};
\node at (4.5,12.75)[font = \small]{$m^1c^2$};
\node at (3.5,12.25)[font = \small]{$\ell^1 m^2$};
\node at (4.5,12.25)[font = \small]{$m^1 \ell^2$};
\node at (3.5,11.75)[font=\small]{$a^1k^2$};
\node at (4.5,11.75)[font=\small]{$k^1a^2$};
\node at (6,9.75)[font = \small]{$a^1a^2$};
\node at (3.5,9.75)[font = \small]{$a^1b^2$};
\node at (4.5,9.75)[font = \small]{$b^1a^2$};
\node at (2,9.75)[font=\small]{$b^1b^2$};
\node at (1.5, 12.5)[font = \small]{$b^1k^2$};
\node at (2.5, 12.5)[font = \small]{$k^1b^2$};
\node at (1.5,12)[font = \small]{$a^1f^2$};
\node at (2.5,12)[font = \small]{$f^1a^2$};
\node at (1.5,13.5)[font = \small]{$c^1c^2$};
\node at (2.5,13.5)[font = \small]{$\ell^1\ell^2$};
\node at (1.5,13)[font = \small]{$c^1 \ell^2$};
\node at (2.5,13)[font = \small]{$\ell^1 c^2$};
\node at (1.5,11.5)[font = \small]{$e^1 m^2$};
\node at (2.5,11.5)[font = \small]{$m^1e^2$};
\node at (1.5,11)[font = \small]{$a^1j^2$};
\node at (2.5,11)[font = \small]{$j^1a^2$};
\end{tikzpicture}
\label{CFK(trefoil) Figure}
\end{figure}

\medskip

As before, the chain complexes in each Alexander grading may be found in Figures Figures \ref{-7/2 Figure}, \ref{-5/2 Figure}, \ref{-3/2 Figure}, \ref{-1/2 Figure}, \ref{1/2 Figure}, \ref{3/2 Figure} and \ref{5/2 Figure}.  Computing the homology of these complexes, we obtain $\widetilde{\mathit{HFK}}(\widetilde{\mathcal D}) = \widehat{\mathit{HFK}}(3_1)\otimes V_1 \otimes W$, whose generators are described in Figure \ref{HFK(trefoil) Figure}.  These elements generate the $E^1$ page of the knot Floer homology spectral sequence of $\widetilde{\mathcal D}$ as a $\mathbb Z_2((q))$-module.  Once again, homology classes which are equivalent under the induced involution $\tau^*$ but have no representative which is invariant under the chain map $\tau^{\#}$ have been included with two equivalent descriptions to emphasize their invariance.

\begin{figure}
\caption{The homology $\widetilde{\mathit{HFK}}(\widetilde{\mathcal D})$, which is  $\widehat{\mathit{HFK}}(3_1)\otimes V_1 \otimes W$.  These elements generate the $E^1$ page of the knot Floer homology spectral sequence of $\widetilde{\mathcal D}$ as a $\mathbb Z_2((q))$-module.}
\label{HFK(trefoil) Figure}
\centering
\begin{tikzpicture}[scale=.9]
\draw(-8,18)edge(6,18);
\draw(-8,18)edge(-8,9);
\draw(6,18)edge(6,9);
\draw(-8,9)edge(6,9);
\draw(-8,18)edge(6,18);
\draw(-7,18)edge(-7,9);
\draw(-6,18)edge(-6,9);
\draw(-8,17)edge(6,17);
\draw (-5,18)edge(-5,9);
\draw(-4,18)edge(-4,9);
\draw(-7,14)edge(6,14);
\draw(-1.5,18)edge(-1.5,9);
\draw(-7,10.5)edge(6,10.5);
\draw(1,18)edge(1,9);
\draw(3.5,18)edge(3.5,9);
\node at (-7.5,17.5){$A_1$};
\node at (-6.5,17.5){$-5$};
\node at (-5.5,17.5){$-4$};
\node at (-4.5,17.5)[font = \small]{$-3$};
\node at (-2.75, 12.25)[font=\small]{$[a^1g^2]=[g^1a^2]$};
\node at (-2.75,17.5){$-2$};
\node at (-2.75,15.5)[font = \small]{$[h^1m^2+m^1h^2]$};
\node at (-.25,17.5) {$-1$};
\node at (2.25,17.5) {$0$};
\node at (4.75,17.5){$1$};
\node at (4.75,12.25)[font = \small]{$[m^1m^2]$};
\node at (-.25,15.75)[font = \small]{$[i^1m^2+m^1e^2]$};
\node at (-.25,15.25)[font = \small]{$[m^1i^2+e^1m^2]$};
\node at (2.25,12.5)[font = \small]{$[\ell^1 m^2+m^1c^2]$};
\node at (2.25,12)[font = \small]{$[m^1 \ell^2 + c^1m^2]$};
\node at (4.75,9.75)[font = \small]{$[a^1a^2]$};
\node at (1.75,9.75)[font = \small]{$[a^1b^2]$};
\node at (2.75,9.75)[font = \small]{$[b^1a^2]$};
\node at (-.8,12.25)[font = \small]{$[a^1f^2]$};
\node at (.3,12.25)[font = \small]{$[f^1a^2]$};
\end{tikzpicture}
\end{figure}

\medskip

The differential $d_1 = (1 + \tau^*)\theta$ on the $E^1$ page of the spectral sequence has the effect of eliminating all elements of $\widetilde{\mathit{HFK}}(\widetilde{\mathcal D})$ which are not invariant under the action of $\tau^*$.  Computing homology with respect to $d_1$ yields the set of generators of Figure \ref{E2 trefoil knot figure}.


\begin{figure}
\caption{Generators for the $E^2$ page of the knot Floer spectral sequence associated to $\widetilde{\mathcal D}$ as a $\mathbb Z_2((q))$ module.}
\label{E2 trefoil knot figure}
\begin{tikzpicture}[scale=.9]
\draw(-8,18)edge(3,18);
\draw(-8,18)edge(-8,9);
\draw(3,18)edge(3,9);
\draw(-8,9)edge(3,9);
\draw(-7,18)edge(-7,9);
\draw(-6,18)edge(-6,9);
\draw(-8,17)edge(3,17);
\draw (-5,18)edge(-5,9);
\draw(-4,18)edge(-4,9);
\draw(-7,14)edge(3,14);
\draw(-1.5,18)edge(-1.5,9);
\draw(-7,10.5)edge(3,10.5);
\draw(-.5,18)edge(-.5,9);
\draw(.5,18)edge(.5,9);
\node at (-7.5,17.5){$A_1$};
\node at (-6.5,17.5){$-5$};
\node at (-5.5,17.5){$-4$};
\node at (-4.5,17.5)[font = \small]{$-3$};
\node at (-2.75, 12.25)[font=\small]{$[a^1g^2]=[g^1a^2]$};
\node at (-2.75,17.5){$-2$};
\node at (-2.75,15.5)[font = \small]{$[h^1m^2+m^1h^2]$};
\node at (-1,17.5) {$-1$};
\node at (0,17.5) {$0$};
\node at (1.75,17.5){$1$};
\node at (1.75,12.25)[font = \small]{$[m^1m^2]$};
\node at (1.75,9.75)[font = \small]{$[a^1a^2]$};
\end{tikzpicture}
\end{figure}

\medskip 

The knot Floer spectral sequence stabilizes on the $E^2$ page in every Alexander grading except $A_1 = -2$.  There is a single nontrivial $d_2$ differential which behaves similarly to the nontrivial $d_2$ differential we earlier saw in the case of the unknot as a doubly-periodic knot.  In particular: consider $d_2([a^1g^2])$.  We compute this differential as follows: first, apply $1+\tau^{\#}$ to $a^1g^2$, obtaining $(a^1g^2 + g^1a^2)\theta$.  Next, we choose an element whose boundary under $\partial_U$ is $(a^1g^2 + g^1a^2)\theta$; one such is $(h^1m^2)\theta$.  Then $d_2([a^1g^2]) = [1+\tau^{\#}(h^1m^2)]\theta) = [h^1m^2 + m^1h^2]\theta^2$. Moreover, we then have $d_2([a^1g^2]\theta^n)=[h^1m^2 + m^1h^2]\theta^{n+2}$ in general. Therefore the Alexander grading $-2$ vanishes on the $E^3$ page of the knot Floer spectral sequence, and the $E^3 = E^{\infty}$ page is isomorphic to $\widetilde{\mathit{HFK}}(\mathcal D)$ after an appropriate shift and rescaling in Alexander $A_1$ gradings.  Generators for this page of the knot Floer spectral sequence appear in Figure \ref{E3(trefoil) Figure}.

\begin{figure}
\caption{Generators for the $E^3 = E^{\infty}$ page of the knot Floer spectral sequence associated to $\mathcal{\widetilde{D}}$.}
\label{E3(trefoil) Figure}
\centering
\begin{tikzpicture}[scale=.9]
\draw(-8,15)edge(1.5,15);
\draw(-8,15)edge(-8,9);
\draw(1.5,15)edge(1.5,9);
\draw(-8,9)edge(1.5,9);
\draw(-7,15)edge(-7,9);
\draw(-6,15)edge(-6,9);
\draw (-5,15)edge(-5,9);
\draw(-4,15)edge(-4,9);
\draw(-7,14)edge(1.5,14);
\draw(-3,15)edge(-3,9);
\draw(-7,10.5)edge(1.5,10.5);
\draw(-2,15)edge(-2,9);
\draw(-1,15)edge(-1,9);
\node at (-7.5,14.5){$A_1$};
\node at (-6.5,14.5){$-5$};
\node at (-5.5,14.5){$-4$};
\node at (-4.5,14.5)[font = \small]{$-3$};
\node at (-3.5,14.5){$-2$};
\node at (-2.5,14.5) {$-1$};
\node at (-1.5,14.5) {$0$};
\node at (.25,14.5){$1$};
\node at (.25,12.25)[font = \small]{$[m^1m^2]$};
\node at (.25,9.75)[font = \small]{$[a^1a^2]$};
\end{tikzpicture}
\end{figure}





\medskip

Notice that for the left-handed trefoil considered as a two-periodic knot, Edmonds' condition is sharp: $g(3_1) = 1 = 2(0) + \frac{3 - 1}{2} = 2g(U) + \frac{\lambda -1}{2}$.  We also see here the realization of Corollary \ref{Fiberedness Corollary}: the left-handed trefoil is fibred, and the highest nontrivial Alexander grading in $\widetilde{\mathit{HFK}}(\widetilde{\mathcal D})$, namely $A_1 = 1$ has rank two.  The two generators in this Alexander grading, $[a_1a_2]$ and $[m_1m_2]$, are preserved over the course of the spectral sequence and become the two generators of $\widetilde{\mathit{HFK}}(\mathcal D)$ in grading $A_1 = 0$ under the isomorphism between the $E^{\infty}$ page of the knot Floer spectral sequence and $\widetilde{\mathit{HFK}}(\mathcal D)$.  Then the highest nontrivial Alexander grading of $\widetilde{\mathit{HFK}}(\mathcal D)$, $A_1 = 0$, also has rank two, corresponding to the unknot being a fibred knot.














\begin{figure}
\caption{The chain complex $\widehat{\mathit{CFL}}(\widetilde{\mathcal D})$ in Alexander grading $A_1 = -\frac{7}{2}$, and the chain complex $\widehat{\mathit{CFK}}(\widetilde{\mathcal D})$ in Alexander grading $A_1 = -5$.  Dashed arrows denote differentials appearing only in the latter.}
\label{-7/2 Figure}
\centering
\begin{tikzpicture}[node distance = 2cm, auto]
\node(1){$h^1h^2$};
\node(2)[below of = 1]{$g^1g^2$};
\path[->,dashed](1)edge(2);
\end{tikzpicture}
\end{figure}


\begin{figure}
\caption{The chain complex $\widehat{\mathit{CFL}}(\widetilde{\mathcal D})$ in Alexander grading $A_1 = -\frac{5}{2}$, and the chain complex $\widehat{\mathit{CFK}}(\widetilde{\mathcal D})$ in Alexander grading $A_1 = -4$.  Dashed arrows denote differentials appearing only in the latter.}
\label{-5/2 Figure}
\centering
\begin{tikzpicture}[node distance = 2cm, auto]
\node(1){$h^1i^2$};
\node(2)[below left = 1.5 cm and .3 cm of 1]{$g^1j^2$};
\node(3)[below right = 1.5 cm and .3 cm of 1]{$f^1g^2$};
\node(4)[below = 3.4 cm of 1]{$e^1h^2$};
\path[->,dashed](1)edge(2);
\path[->,dashed](1)edge(3);
\path[->](2)edge(4);
\path[->](3)edge(4);
\node(5)[right of = 1, node distance = 5 cm]{$i^1h^2$};
\node(6)[below left = 1.5 cm and .3 cm of 5]{$g^1f^2$};
\node(7)[below right = 1.5 cm and .3 cm of 5]{$j^1g^2$};
\node(8)[below = 3.4 cm of 5]{$h^1e^2$};
\path[->,dashed](5)edge(6);
\path[->,dashed](5)edge(7);
\path[->](6)edge(8);
\path[->](7)edge(8);
\end{tikzpicture}
\end{figure}

\begin{figure}
\caption{The chain complex $\widehat{\mathit{CFL}}(\widetilde{\mathcal D})$ in Alexander grading $A_1 = -\frac{3}{2}$, and the chain complex $\widehat{\mathit{CFK}}(\widetilde{\mathcal D})$ in Alexander grading $A_1 = -3$.  Dashed arrows denote differentials appearing only in the latter.}
\label{-3/2 Figure}
\centering
\begin{tikzpicture}[node distance = 2 cm, auto]
\node(1){$i^1i^2$};
\node(2)[below left = 1.5 cm and .5 cm of 1]{$k^1g^2$};
\node(3)[below right = 1.5 cm and .5 cm of 1]{$g^1k^2$};
\node(4)[left of = 2]{$f^1f^2$};
\node(5)[right of = 3]{$j^1j^2$};
\path[->, dashed](1)edge(2);
\path[->, dashed](1)edge(3);
\path[->, dashed](1)edge(4);
\path[->, dashed](1)edge(5);
\node(6)[below = 3.5 cm of 1]{$e^1e^2$};
\node(7)[left of = 6]{$c^1h^2$};
\node(9)[left of = 7]{$h^1c^2$};
\node(8)[right of = 6]{$\ell^1h^2$};
\node(10)[right of = 8]{$h^1\ell^2$};
\path[->, dashed](4)edge(6);
\path[->](4)edge(9);
\path[->](4)edge(7);
\path[->, dashed](5)edge(6);
\path[->](5)edge(8);
\path[->](5)edge(10);
\path[->](2)edge(9);
\path[->](2)edge(8);
\path[->](3)edge(7);
\path[->](3)edge(10);
\node(11)[below = 3.5 cm of 2]{$g^1b^2$};
\node(12)[below = 3.5 cm of 3]{$b^1g^2$};
\path[->](6)edge(11);
\path[->](6)edge(12);
\path[->, dashed](9)edge(11);
\path[->, dashed](7)edge(12);
\path[->, dashed](8)edge(11);
\path[->, dashed](10)edge(12);
\node(13)[below of = 11]{$e^1i^2$};
\node(14)[below of = 12]{$i^1e^2$};
\node(15)[below of = 13]{$f^1j^2$};
\node(16)[below of = 14]{$j^1f^2$};
\path[->](13)edge(15);
\path[->](14)edge(16);
\end{tikzpicture}
\end{figure}


\begin{figure}
\caption{The chain complex $\widehat{\mathit{CFL}}(\widetilde{\mathcal D})$ in Alexander grading $A_1 = -\frac{1}{2}$, and the chain complex $\widehat{\mathit{CFK}}(\widetilde{\mathcal D})$ in Alexander grading $A_1 = -2$.  Dashed arrows denote differentials appearing only in the latter.}
\label{-1/2 Figure}
\centering
\rotatebox{90}{
\begin{tikzpicture}[node distance = 2 cm, auto]
\node(1){$k^1j^2$};
\node(2)[right of = 1, node distance = 4 cm]{$f^1k^2$};
\node(3)[right of = 2, node distance = 4 cm]{$k^1f^2$};
\node(4)[right of = 3, node distance = 4 cm]{$j^1k^2$};
\node(9)[below left = 3 cm and .02 cm of 2]{$h^1m^2$};
\node(12)[below right = 3 cm and .02 cm of 3]{$m^1h^2$};
\node(7)[right of = 9]{$\ell^1e^2$};
\node(8)[left of = 12]{$e^1\ell^2$};
\node(11)[left of = 9]{$\ell^1i^2$};
\node(5)[left of = 11]{$c^1i^2$};
\node(10)[left of = 5]{$e^1c^2$};
\node(14)[right of = 12]{$i^1\ell^2$};
\node(6)[right of = 14]{$i^1c^2$};
\node(13)[right of = 6]{$c^1e^2$};
\path[->](1)edge(11);
\path[->, dashed](1)edge(10);
\path[->](1)edge(9);
\path[->](2)edge(9);
\path[->](2)edge(5);
\path[->, dashed](2)edge(8);
\path[->, dashed](3)edge(7);
\path[->](3)edge(6);
\path[->](3)edge(12);
\path[->](4)edge(12);
\path[->,dashed](4)edge(13);
\path[->](4)edge(14);
\node(15)[below right = 7 cm and .05 cm of 2]{$b^1j^2$};
\node(16)[below left = 7 cm and .05 cm of 3]{$j^1b^2$};
\node(17)[left of = 15, node distance = 2.2 cm]{$a^1g^2$};
\node(18)[left of = 17, node distance = 2.2 cm]{$f^1b^2$};
\node(19)[right of = 16, node distance = 2.2 cm]{$g^1a^2$};
\node(20)[right of = 19, node distance = 2.2 cm]{$b^1f^2$};
\path[->, dashed](11)edge(18);
\path[->, dashed](11)edge(19);
\path[->](10)edge(18);
\path[->](10)edge(17);
\path[->, dashed](9)edge(17);
\path[->, dashed](9)edge(19);
\path[->, dashed](5)edge(17);
\path[->, dashed](5)edge(15);
\path[->](7)edge(17);
\path[->](7)edge(16);
\path[->](8)edge(15);
\path[->](8)edge(19);
\path[->, dashed](6)edge(16);
\path[->, dashed](6)edge(19);
\path[->, dashed](12)edge(17);
\path[->, dashed](12)edge(19);
\path[->, dashed](14)edge(17);
\path[->, dashed] (14)edge(20);
\path[->](13)edge(20);
\path[->](13)edge(19);
\end{tikzpicture}}
\end{figure}

\begin{figure}
\caption{The chain complex $\widehat{\mathit{CFL}}(\widetilde{\mathcal D})$ in Alexander grading $A_1 = \frac{1}{2}$, and the chain complex $\widehat{\mathit{CFK}}(\widetilde{\mathcal D})$ in Alexander grading $A_1 = -1$.  Dashed arrows denote differentials appearing only in the latter.}
\label{1/2 Figure}
\centering
\begin{tikzpicture}[node distance = 2 cm, auto]
\node(1){$k^1k^2$};
\node(2)[below left = 1.5 cm and .5 cm of 1]{$i^1m^2$};
\node(3)[below right = 1.5 cm and .5 cm of 1]{$m^1i^2$};
\node(4)[left of = 2]{$e^1m^2$};
\node(5)[left of = 4]{$c^1c^2$};
\node(6)[right of = 3]{$m^1e^2$};
\node(7)[right of = 6]{$\ell^1\ell^2$};
\path[->](1)edge(2);
\path[->](1)edge(3);
\path[->, dashed](1)edge(5);
\path[->, dashed](1)edge(7);
\node(8)[below = 4.2 cm of 1]{$b^1b^2$};
\node(9)[left of = 8]{$a^1f^2$};
\node(10)[left of = 9]{$f^1a^2$};
\node(11)[right of = 8]{$a^1j^2$};
\node(12)[right of = 11]{$j^1a^2$};
\path[->, dashed](5)edge(8);
\path[->](5)edge(9);
\path[->](5)edge(10);
\path[->, dashed](7)edge(8);
\path[->](7)edge(11);
\path[->](7)edge(12);
\path[->](4)edge(10);
\path[->](4)edge(11);
\path[->](6)edge(9);
\path[->](6)edge(12);
\path[->, dashed](2)edge(9);
\path[->, dashed](2)edge(12);
\path[->, dashed](3)edge(10);
\path[->, dashed](3)edge(11);
\node(13)[below = 4 cm of 2]{$c^1\ell^2$};
\node(14)[below = 4 cm of 3]{$\ell^1c^2$};
\node(15)[below of = 13]{$b^1k^2$};
\node(16)[below of = 14]{$k^1b^2$};
\path[->](13)edge(15);
\path[->](14)edge(16);
\end{tikzpicture}
\end{figure}

\begin{figure}
\caption{The chain complex $\widehat{\mathit{CFL}}(\widetilde{\mathcal D})$ in Alexander grading $A_1 = \frac{3}{2}$, and the chain complex $\widehat{\mathit{CFK}}(\widetilde{\mathcal D})$ in Alexander grading $A_1 = 0$.  Dashed arrows denote differentials appearing only in the latter.}
\label{3/2 Figure}
\centering
\begin{tikzpicture}[node distance = 2cm, auto]
\node(1){$\ell^1m^2$};
\node(2)[right of = 1] {$m^1c^2$};
\node(3)[right of = 2, node distance = 3 cm] {$c^1m^2$};
\node(4)[right of = 3]{$m^1\ell^2$};
\node(5)[below of = 1]{$a^1b^2$};
\node(6)[below of = 2]{$k^1a^2$};
\node(7)[below of = 3]{$a^1k^2$};
\node(8)[below of = 4]{$b^1a^2$};
\path[->, dashed](1)edge(5);
\path[->](2)edge(6);
\path[->](3)edge(7);
\path[->, dashed](4)edge(8);
\path[->,](1)edge(6);
\path[->, dashed](2)edge(5);
\path[->, dashed](3)edge(8);
\path[->,](4)edge(7);
\end{tikzpicture}
\end{figure}

\begin{figure}
\caption{The chain complex $\widehat{\mathit{CFL}}(\widetilde{\mathcal D})$ in Alexander grading $A_1 = \frac{5}{2}$, and the chain complex $\widehat{\mathit{CFK}}(\widetilde{\mathcal D})$ in Alexander grading $A_1 = 1$.  Dashed arrows denote differentials appearing only in the latter.}
\label{5/2 Figure}
\centering
\begin{tikzpicture}[node distance = 2 cm, auto]
\node(1){$m^1m^2$};
\node(2)[below of = 1]{$a^1a^2$};
\path[->, dashed][bend left](1)edge(2);
\path[->, dashed][bend right](1)edge(2);
\end{tikzpicture}
\end{figure}



\raggedright

\bibliographystyle{amsplain}
\bibliography{bibliography1}

\end{document}
PeriodicKnotsPaperHendricks.bbl
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
% \MRhref is called by the amsart/book/proc definition of \MR.
\providecommand{\MRhref}[2]{%
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}
\providecommand{\href}[2]{#2}
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