Non-loose Legendrian spheres with trivial Contact Homology DGA
Tobias Ekholm
Loose Legendrian $n$-submanifolds, $n\ge 2$, were introduced in \cite{Murphyloose and proved to be flexible in the h-principle sense: any two loose Legendrian submanifolds that are formally Legendrian isotopic are also actually Legendrian isotopic. Legendrian contact homology is a Floer theoretic invariant that associates a differential graded algebra (DGA) to a Legendrian submanifold. The DGA of a loose Legendrian submanifold is trivial. We show that the converse is not true by constructing non-loose Legendrian $n$-spheres in standard contact $(2n+1)$-space, $n\ge 2$, with trivial DGA.
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\begin{document}

\title[Non-loose Legendrian spheres\dots]
{Non-loose Legendrian spheres with trivial Contact Homology DGA}

\author{Tobias Ekholm}
\address{Uppsala University, Box 480, 751 06 Uppsala, Sweden\newline
\indent Insitute Mittag-Leffler, Aurav 17, 182 60 Djursholm, Sweden}
\email{tobias.ekholm@math.uu.se}
\thanks{The author was funded by the Knut and Alice Wallenberg Foundation and by the Swedish Research Council, 2012-2365.}


\begin{abstract}
Loose Legendrian $n$-submanifolds, $n\ge 2$, were introduced in \cite{Murphyloose} and proved to be flexible in the h-principle sense: any two loose Legendrian submanifolds that are formally Legendrian isotopic are also actually Legendrian isotopic. Legendrian contact homology is a Floer theoretic invariant that associates a differential graded algebra (DGA) to a Legendrian submanifold. The DGA of a loose Legendrian submanifold is trivial. We show that the converse is not true by constructing non-loose Legendrian $n$-spheres in standard contact $(2n+1)$-space, $n\ge 2$, with trivial DGA.  
\end{abstract}

\maketitle

\section{Introduction}
Let $Y$ be a contact $(2n+1)$-manifold and $\Lambda\subset Y$ a Legendrian submanifold. Let $\R^{2n+1}_{\st}$ denote $\R^{2n+1}$ with the standard contact structure given by the kernel of the contact 1-form, $dz-y\cdot dx$, where $y\cdot dx=\sum_{j=1}^{n}y_jdx_j$ for standard coordinates $(x,y)=(x_1,y_1,\dots,x_n,y_n)$ on $\R^{2n}$ and where $z$ is a linear coordinate on $\R$.   
If $q\in Y$ then Darboux's theorem shows that $q$ has a neighborhood in $Y$ contactomorphic to a neighborhood of the origin in $\R^{2n+1}_{\st}$. If furthermore $q\in\Lambda$ then the Darboux chart can be chosen so that $\Lambda$ corresponds to the intersection of the neighborhood of the origin with the subspace $\{y=0,z=0\}$. 

Loose Legendrian submanifolds were introduced by Murphy in \cite{Murphyloose} and are characterized by the following local model. A \emph{loose chart} $(B_{0},\Delta_0)$ is a topological ball $B_0$ around the origin in $\R^{2n+1}_{\st}$ with a certain embedded Legendrian disk $\Delta_{0}\subset B_{0}$ defined as follows.
Let $\gamma$ be the Legendrian arc in $U_{a}\subset \R^{3}_{\st}$ shown in Figure \ref{fig:loosechart}, where 
\[ 
U_{a}=\left\{(x_0,y_0,z)\in\R^{3}\colon -\tfrac12 a < z < \tfrac12 a\right\}
\]
and let
\begin{align*} 
D_{\rho} &= \left\{(x,y)\in\R^{2n-2}\colon |x|\le \rho,\;|y|\le \rho\right\},\\
\delta_{\rho} &=  \left\{(x,y)\in\R^{2n-2}\colon |x|\le \rho,\;y=0\right\}\subset D_\rho
\end{align*} 
then 
\[ 
(B_{0},\Delta_{0})= \left(U_{a}\times D_{\rho},\gamma\times \delta_{\rho}\right)\subset \R^{2n+1}_{\st}, \quad a<\tfrac12 \rho^{2}.
\]
\begin{figure}[ht]
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=.75\linewidth]{loosefront}
\end{subfigure}%
\begin{subfigure}{.5\textwidth}
\centering
\includegraphics[width=.75\linewidth]{frontkink}
\end{subfigure}
\caption{Left: the Legendrian arc $\gamma$. Right: the front of $\Delta_0$ in a loose chart $B_0$}
\label{fig:loosechart}
\end{figure}
A Legendrian submanifold $\Lambda\subset Y$ is called \emph{loose} if there is a point $q\in\Lambda$ and a loose chart $\psi\colon (B_0,\Delta_{0})\to (Y,\Lambda)$ where $\psi$ is a contact embedding with $\psi(0)=q$, see Figure \ref{fig:loosechart}. 

Murphy showed that loose Legendrian submanifolds are flexible in the h-principle sense \cite[Theorem 1.2]{Murphyloose}, meaning the following. Two Legendrian submanifolds are called \emph{formally} Legendrian isotopic if they are isotopic as smooth submanifolds through an isotopy that can be covered by a homotopy of injective linear maps from the tangent spaces of the submanifold to the tangent spaces of the ambient contact manifold such that the image is everywhere an isotropic subspace. The flexibility result then states that two loose Legendrian submanifolds that are formally isotopic are also actually Legendrian isotopic, i.e.~there is a continuous 1-parameter family of Legendrian submanifolds connecting them. 

Legendrian contact homology is a Floer homological theory associated to Legendrian submanifolds. It is a part of Symplectic Field Theory \cite{EGH} and is defined as follows. To a Legendrian submanifold $\Lambda\subset Y$ one associates a differential graded algebra (DGA) $\Aa_{\Lambda}$. The generators of the algebra are Reeb chords, i.e.~flow lines of the Reeb vector field determined by the contact form on $Y$, that begins and ends on $\Lambda$, and closed Reeb orbits. The differential of the DGA counts holomorphic curves in the symplectization $\R\times Y$ of $Y$ with Lagrangian boundary condition $\R\times\Lambda$. These curves are either disks with one positive boundary puncture where the disk is asymptotic to a Reeb chord and with several negative boundary and interior punctures where the disk is asymptotic to chords and orbits, respectively, or they are spheres with one positive and several negative interior punctures with orbit asymptotics.    
 
In the present paper we will consider DGAs only for Legendrian submanifolds of $\R^{2n+1}_{\st}$. Here the situation simplifies since there are no closed orbits and since a generic compact Legendrian submanifold has only finitely many chords. The foundations of the theory in this situation were worked out in \cite{EES_R2n+1, EES, EES_orientation}. Furthermore, in this case the rigid holomorphic disks of the differential admit descriptions in terms of Morse flow trees which make the DGA rather computable, see \cite{E}. 

It is not hard to show that the DGA of a loose Legendrian submanifold is trivial: in the loose chart one finds after isotopy an arbitrarily short Reeb chord $c$ with $dc=1$, see~\cite[Proposition 4.8]{EES_noniso}, and then multiplication by $c$ gives a chain homotopy connecting the identity to the zero map, see Lemma \ref{l:dga}. In view of this it is natural to ask whether triviality of the DGA of a Legendrian submanifold implies looseness. Our main result shows that this is not the case. More precisely we have the following.

\begin{thm}\label{t:main}
For every $n\ge 2$, there is a Legendrian $n$-sphere $\Lambda\subset \R^{2n+1}_{\mathrm{st}}$ such that the Legendrian DGA of $\Lambda$ is trivial but $\Lambda$ is not loose.
\end{thm}      

Loose Legendrian submanifolds can also be characterized by being stabilizations of other Legendrian submanifolds, see \cite[Proposition 2.5--6]{Murphyloose}. From this point of view, Theorem \ref{t:main} should be compared to \cite{Sivek} which shows that there are Legendrian knots in $\R^{3}_{\st}$ with trivial DGA that are not stabilizations.    

We prove Theorem \ref{t:main} as follows. First, elaborating on results from \cite{BC,CNS}, we find a Legendrian $(n-1)$-sphere $\Sigma\subset \R^{2n-1}_{\mathrm{st}}$ that bounds two distinct Lagrangian disks $W_0$ and $W_1$ in the symplectization $\R\times\R^{2n-1}_{\mathrm{st}}$. 
More concretely this means that $W_j$, $j=0,1$ are Lagrangian disks that agree with the cylinder $\R\times \Sigma$ in $[T,\infty)\times \R^{2n+1}_{\st}$ for some $T>0$. Such Lagrangian fillings induce chain maps $\Aa_{\Sigma}\to\Z$ called augmentations and the augmentations of $W_0$ and $W_1$ are different. 

Using a symplectomorphism $\phi$ between $[T,-\infty)\times\R^{2n-1}_{\mathrm{st}}$ and the half-space $[0,\infty)\times\R^{2n-1}$ of standard symplectic $2n$-space, $\R^{2n}_{\st}$, with symplectic form $dx\wedge dy=\sum_{j=1}^{n} dx_j\wedge dy_j$, we construct from the Lagrangian disk $W_j$, $j=0,1$ a Morse cobordism, as in \cite[Section 2.3]{EHK}. This is an immersed Lagrangian submanifold $W_{j}^{\Mo}$ with boundary, where $W_j^{\Mo}$ lies in  $[0,\infty)\times\R^{2n-1}$ and has its boundary in $0\times\R^{2n-1}$. Here $W_{j}^{\Mo}$ agrees with $\phi(W_j)$ outside an $\epsilon$-neighborhood of $0\times \R^{2n-1}$, its interior is an embedded submanifold of $(0,\infty)\times\R^{2n-1}$, and along its boundary it lies in $0\times 0\times \R^{2n-2}$ and equals the projection of $\Sigma$ into $\R^{2n-2}$. Consequently the double points of $W_{j}^{\Mo}$ lie along its boundary and are in natural one to one correspondence with the Reeb chords of $\Sigma$. 

The image of $W^{\Mo}_{j}$ under a reflection in the first two coordinates of $\R^{2n}_{\st}$ is again an exact Lagrangian submanifold lying in the opposite half space but agreeing with $W_{j}^{\Mo}$ along the boundary. Gluing two such Lagrangian disks $W_i^{\Mo}$ and $W_j^{\Mo}$ lying in two different half spaces we construct an immersed Lagrangian sphere $S$ with double points corresponding to the Reeb chords of $\Sigma$.  The Legendrian sphere $\Lambda$ in Theorem \ref{t:main} is the Legendrian lift of such an immersed Lagrangian sphere $S$ constructed by gluing $W_0^{\Mo}$ and $W^{\Mo}_{1}$. We also consider the Legendrian spheres $\Lambda_{j}$, $j=0,1$ constructed exactly as $\Lambda$ but using instead the same filling $W_{j}^{\Mo}$ on both sides.   
 
To prove that $\Lambda$ fulfills the requirements of Theorem \ref{t:main}, we calculate the DGAs of $\Lambda$ and of $\Lambda_j$, $j=0,1$, using flow tree methods developed in \cite{E,EK}. The computations show that $\Lambda$ has trivial DGA whereas $\Lambda_j$, $j=0,1$ has non-trivial DGA. We then assume that there is a loose chart for the sphere $\Lambda$. By employing contact lifts of three Hamiltonian isotopies of $\R^{2n}_{\st}$, we show that the loose chart can be moved sufficiently far away from the intersection of $\Lambda$ with an $\epsilon$-neighborhood of one of the half spaces of $\R^{2n}_{\st}$ times $\R$, that we can alter one of the halves $W^{\Mo}_{i}$ of $\Lambda$ by cut and paste in the complement of the loose chart, changing it to a copy of $W^{\Mo}_{j}$, $i\ne j$. This would then give a loose chart for $\Lambda_j$ which contradicts our DGA calculation. 
 
The construction used in the proof of Theorem \ref{t:main} has independent interest. For example, given any Legendrian submanifold $\Sigma$ of $\R^{2n-1}_{\st}$ that admits an exact Lagrangian filling $W$ in the symplectization $\R\times\R^{2n-1}_{\st}$, it produces a Legendrian embedding of the double of $W$ into $\R^{2n+1}_{\st}$ with Reeb chords in natural one to one correspondence with the Reeb chords of $\Sigma$ and with DGA where all holomorphic disks contributing to the differential have at most one negative puncture and the DGA itself is simply the tensor algebra on the singular homology of $W$. This thus gives a large class of Legendrian submanifolds with simple DGAs and it would be interesting to understand how their Legendrian isotopy classes depend on $\Sigma$ and $W$. 

The paper is organized as follows. In Section \ref{S:LegLag} we construct Legendrian $(n-1)$-spheres with two distinct Lagrangian disk fillings and compute their augmentation maps and the DGAs of the Legendrian $n$-spheres constructed from the fillings as described above. In Section \ref{S:nloose} we then prove Theorem \ref{t:main}. 

\subsection*{Acknowledgments} The author thanks an anonymous referee for helpful comments that improved Sections \ref{s:morsedim2} and \ref{s:morsedim+}. 

\section{Legendrian knots and Lagrangian fillings}\label{S:LegLag}
In this section we construct Legendrian submanifolds from Lagrangian cobordisms and compute their associated DGAs. We first study the case of Legendrian $1$-knots and Lagrangian surfaces and then explain how to transport the results obtained into higher dimension. 

\subsection{Gluing Morse cobordisms of dimension 2}\label{s:gludim2}
Let $K\subset\R^{3}_{\mathrm{st}}$ be a Legendrian knot that bounds an exact Lagrangian surface $F$ in the symplectization $\R\times\R^{3}_{\st}$ of $\R^{3}_{\st}$. In \cite{EHK}, a \emph{Morse cobordism} $F^{\mathrm{Mo}}$ in $\R^{4}_{\st}$ was associated to $F$ as follows. Let $(x_1,y_1,x_2,y_2)$ be coordinates on $\R^{4}_{\st}$ with the standard symplectic form $-d(y_1dx_1+y_2dx_2)$ and recall that the symplectic form on the symplectization $\R\times\R^{3}_{\st}$ is $d(e^{t}\alpha)$, where $\alpha$ is the contact form $dz-ydx$ on $\R^{3}_{\st}$. For fixed $T>0$, the maps $\phi_{T}\colon \R\times\R^{3}_{\st}\to\R^{4}_{\st}$,
\[ 
\phi_T(x,y,z,t) = (x,e^{t}y, e^{t}, z) - (0,0,e^{T}+1,0)
\]
are exact symplectomorphisms that take the slice $\{t=T\}$ in the symplectization to the hyperplane $\{x_2=-1\}$ in $\R^{4}_{\st}$. Furthermore, if the Lagrangian surface $F$ is a cylinder on a Legendrian knot $K$ around the slice $t=T$ then its image is a conical Lagrangian surface in $\R^{4}_{\st}$, which means that its front, i.e.~its projection to the subspace $\{y=0\}$ which generically is a singular surface that determines the Lagrangian surface via the equations $y_{j}=\frac{\partial z}{\partial x_j}$, is the front of the Legendrian knot in the $(x_1,z)$-plane scaled by a monotonically increasing function $h(x_2)=x_2+(e^{T}+1)$ near $x_2=-1$. Write $F_{T}=((-\infty,T]\times\R^{3}_{\st})\cap F$ and assume that $F$ is cylindrical on the knot $K$ for $t>T-1$. The Morse cobordism $F^{\Mo}$ is now constructed by continuing the front of $\phi_{T}(F_{T})$ as a cylinder on the front of $K$ scaled by a monotone function $h(x_2)$ extending $x_2-(e^{T}+1)$ over $-1\le x_2\le 0$ and having a Morse maximum at $x_2=0$, see Figure \ref{fig:morsecob} and \cite[Section 2.3]{EHK} for more details.
\begin{figure}[ht]
\centering
\includegraphics[width=.6\linewidth]{morsecob}
\caption{Schematic picture of the front of a Morse cobordism.}
\label{fig:morsecob}
\end{figure}

    
Note that the only double points of $F^{\Mo}$ lies in $\{x_{2}=0\}$ and correspond naturally to the Reeb chords of $K$ and that $y_2=0$ along $F^{\mathrm{Mo}}\cap \{x_{2}=0\}$. 

Consider the image of $F^{\mathrm{Mo}}$ under the reflection 
\[ 
(x_1,y_1,x_2,y_2)\mapsto (x_{1},y_{1},-x_{2},-y_{2})
\]    
and note that this is also an immersed Lagrangian surface lying in the half space of $\R^{4}_{\st}$ where $x_2\ge 0$. We write $F^{\mathrm{Mo}+}$ for this surface and $F^{\mathrm{Mo}-}$ for the original Morse cobordism, where the sign indicates the sign of the $x_2$-coordinate in the half space where the Lagrangian surface lies. 

Consider two exact Lagrangian fillings $F$ and $G$ of $K$. Since the two corresponding Morse cobordisms agree near $\{x_2=0\}$ the two Lagrangian surfaces $F^{\Mo-}$ and $G^{\Mo+}$ glue together to a smooth exact immersed Lagrangian surface $F^{\Mo-}\,\cup_{\partial}\,  G^{\Mo+}$ which topologically is obtained by gluing the two surfaces along their common boundary corresponding to $K$. We denote the Legendrian lift of $F^{\Mo-}\,\cup_{\partial}\, G^{\Mo+}$ into $\R^{5}_{\st}$ by $\Lambda(F,G)$. 


\subsection{Background on DGAs and cobordism maps}
We give a brief description of the contact homology algebras that we will use, referring to \cite{EES_R2n+1, EES} for details.
Let $\Lambda\subset \R^{2n+1}_{\st}$ be a Legendrian submanifold. We write $\Aa_{\Lambda}$ for the DGA of $\Lambda$.  The generators of the algebra $\Aa_{\Lambda}$ are the Reeb chords of $\Lambda$, which in $\R^{2n+1}_{\st}$ are simply line segments parallel to the $z$-axis that begin and end on $\Lambda$. We assume that $\Lambda$ is generic so that there are only finitely many Reeb chords. Furthermore we assume that the Maslov class, see \eqref{eq:Maslovdef}, of $\Lambda$ vanishes in which case the algebra $\Aa_{\Lambda}$ has an integer grading where generators $c$ are graded by a Maslov index $|c|$. In this paper our main concern is Legendrian spheres in higher dimension and we will consider the algebra $\Aa_{\Lambda}$ as an algebra over $\Z$. For Legendrian submanifolds $\Lambda'$ of $\R^{2n+1}_{\st}$ with non-trivial first homology group $H_1(\Lambda')\ne 0$, there are more elaborate versions of the algebra, with coefficients in the group ring of $H_2(\R^{2n+1},\Lambda)\approx H_{1}(\Lambda)$, see e.g.~\cite{EENS}. For spheres and other manifolds $\Lambda$ with $H_1(\Lambda)=0$, the more elaborate algebra reduces to the algebra $\Aa_{\Lambda}$ over $\Z$ considered here. 

The differential in $\Aa_{\Lambda}$ is defined through a holomorphic curve count in the symplectization of $\R^{2n+1}_{\st}$, which is the exact symplectic manifold $\R\times\R^{2n+1}_{\st}$ with symplectic form $d(e^{t}\alpha)$, where $t$ is a coordinate in the $\R$-factor and where $\alpha=dz-y\cdot dx$ is the contact form on $\R^{2n+1}_{\st}$, as follows. If $a$ is a Reeb chord of $\Lambda$ then
\[ 
d a = \sum_{|a|-|\mathbf{b}|=1} |\mathcal{M}(a;\mathbf{b})/\R|\mathbf{b},
\]   
where the notation is as follows:
\begin{itemize}
\item $\mathbf{b}=b_1\dots b_m$ ranges over all Reeb chord monomials;
\item $\mathcal{M}(a;\mathbf{b})$ denotes the oriented moduli space of holomorphic disks in the symplectization $\R\times\R^{2n+1}_{\st}$ with Lagrangian boundary condition $\R\times\Lambda$ and with the following characteristics:
\begin{itemize}
\item one positive puncture where the disk is asymptotic to the Reeb chord strip $\R\times a$ at $+\infty$;
\item $m$ negative punctures where the disk is asymptotic to $\R\times b_j$ at $-\infty$, for $j=1,\dots,m$, in the order determined by the boundary orientation of the disk. 
\end{itemize}
\end{itemize}
These disks are holomorphic with respect to an almost complex structure on $\R\times\R^{2n+1}_{\st}$ that splits as a complex structure on the contact planes compatible with the exterior derivative of the contact form and a complex structure that intertwines the $\R$-direction and the Reeb vector field. With such an almost complex structure the holomorphic disks can be translated along the $\R$-factor and the formal dimension of a moduli space as described is 
\[ 
\dim(\mathcal{M}(a;\mathbf{b}))=|a|-|\mathbf{b}|=|a|-\sum_{j=1}^{m}|b_j|.
\] 
By compactness properties of holomorphic disks, see \cite{BEHWZ}, the sum in the definition of the differential is for generic data a finite sum of points with signs. The orientation of the moduli spaces is induced from a gluing compatible system of orientations on determinant bundles over configuration spaces for holomorphic disks, that in turn requires a choice of a spin structure on $\Lambda$, see \cite{EES_orientation}.  

\begin{rmk}
For the class of almost complex structures on the contact planes in $\R^{2n+1}_{\st}$ that are invariant under the Reeb flow, there is a straightforward relation between the moduli spaces of holomorphic disks in the symplectization as described above and the moduli spaces of their projections to $\R^{2n}_{\st}$, see \cite{D-R}. In \cite{EES, EES_R2n+1}, the whole theory was described in terms of projected disks.  
\end{rmk}



Consider next an exact Lagrangian submanifold $L$ in $\R\times\R^{2n+1}_{\st}$. Recall that the Maslov class is the integer valued map on 
\begin{equation}\label{eq:Maslovdef}
H_2(\R\times\R^{2n+1}_{\st},L)\approx H_1(L)
\end{equation}
that measures the obstruction to extending the field of Lagrangian tangent planes $\gamma^{\ast}(TL)$ along an oriented loop $\gamma$ in $L$ as a field of Lagrangian planes in $\Sigma^{\ast}(T\,\R\times\R^{2n+1}_{\st})$ over an oriented surface $\Sigma$ in $\R\times\R^{2n+1}_{\st}$ with $\partial \Sigma=\gamma$. We consider Lagrangian submanifolds $L$ of Maslov class $0$ that agrees with the Lagrangian cone $\R\times \Lambda$ in  $\{t>T\}$, for some $T>0$. We say that such an $L$ is an \emph{exact Lagrangian filling} of $\Lambda$. (Our main concern below are Lagrangian fillings with the topology of the disk, for which the Maslov class $0$ condition is automatic.) 

If $L$ is an exact Lagrangian filling of $\Lambda$ then there is a chain map $\Phi_{L}\colon \mathcal{A}_{\Lambda}\to\Z$, where $\Z$ lies in degree $0$ and has the trivial differential, see e.g.~\cite{EHK},  
\begin{equation}\label{eq:cobchmap} 
\Phi_{L}\colon \mathcal{A}_{\Lambda}\to\Z.
\end{equation}
The chain map $\Phi_{L}$ is defined on generators as follows:
\begin{equation}\label{eq:cobchmapdiskcount} 
\Phi_{L}(a)=\sum_{|a|=0} |\mathcal{M}(a)|,
\end{equation}
where $\mathcal{M}(a)$ denotes the oriented moduli space of holomorphic disks with positive puncture at the Reeb chord $a$ and with boundary on $L$, which, under the condition $|a|=0$, is a compact oriented $0$-manifold for generic data. 

\begin{rmk}
In \cite{EHK} chain maps $\Phi_{L}$ as in \eqref{eq:cobchmap}, defined using moduli spaces of holomorphic disks as in \eqref{eq:cobchmapdiskcount}, were studied only as maps over $\Z_{2}$. Here we will mainly be interested in the holomorphic disks that correspond to the disks in \eqref{eq:cobchmapdiskcount} but which has boundary on the Morse cobordism $L^{\Mo}$ associated to $L$, see Sections \ref{s:morsedim2} and \ref{s:morsedim+}. For the Lagrangian boundary condition given by $L^{\Mo}$, orientations of moduli spaces of disks with one positive puncture are defined in \cite{EES_orientation} and we will use the corresponding signs in the cobordism map.  
\end{rmk}

\subsection{The DGA of glued Morse cobordisms in dimension 2}\label{s:morsedim2}
Let $K$ be a Legendrian knot in $\R^{3}_{\st}$ and let $F$ be an oriented exact  Lagrangian filling of $K$. In \cite{EHK} it was shown that there is a natural one to one correspondence between rigid holomorphic disks on $F$ (for a certain almost complex structure) and rigid Morse flow trees determined by $F^{\Mo}$. (More precisely, this follows from \cite[Lemmas 5.2, 5.11, and 5.12]{EHK}.) We give a brief description of flow trees in this setting and refer to \cite[Sections 2.2 and 3]{E} or \cite[Section 4]{EHK} for details. 

Recall that $F^{\Mo}$ is an exact Lagrangian submanifold in $\R^{4}_{\st}$. If $z\colon F^{\Mo}\to\R$ satisfies $dz=y_1dx_1+y_2dx_2$, then $z$ is unique up to a constant shift and determines a Legendrian lift of $F^{\Mo}$ into $\R^{5}_{\st}$. The projection of this lift to $\R^{2}\times\R$ with coordinates $(x_1,x_2,z)$ is the front projection of $F^{\Mo}$. For generic exact Lagrangian submanifolds the front has cusp edge singularities of codimension one and swallow tail singularties of codimenion two and no other singularities. Outside the singularities, the front consists of graphs of locally defined functions. We will consider differences of pairs of such local functions and call them local function differences. 
Then self intersection points of $F^{\Mo}$ correspond to critical points of local function differences and at regular points the local function differences define local gradients. A Morse flow tree of $F^{\Mo}$ is a continuous map of a tree $\Gamma$ into $\R^{2}$ such that any edge of $\Gamma$ is a flow segment of a local gradient of the front, and such that the natural ``oriented double lift'' that lifts the flow segments to the sheets of $F^{\Mo}$ defining the local functions is a closed curve. A rigid flow tree is a flow tree of formal dimension zero that is transversely cut out. Such a tree has only 1-, 2-, and 3-valent vertices of specific characters, see \cite[Section 3]{E} or \cite[Section 4.3]{EHK}.        

Assume that $K$ has two Lagrangian fillings $F$ and $G$ and recall the construction of the 2-dimensional Legendrian submanifold $\Lambda(F,G)$ obtained by lifting the immersed exact Lagrangian obtained from gluing $F^{\Mo}$ and $G^{\Mo}$ along their common boundary, see Section \ref{s:gludim2}.

We next compute the DGA $\Aa_{\Lambda(F,G)}$ in terms of the DGA $\Aa_{K}$ of $K$ and the cobordism maps $\Phi_{F}$ and $\Phi_{G}$ induced by $F$ and $G$, respectively. In other words, we find the Reeb chord generators and their gradings and then compute the differential of any generator. The description of the generators is straightforward. The computation of the differential is more involved but follows more or less directly by combining results from \cite{EK} and \cite{EHK}, see the proof of Lemma \ref{l:dga} for details.


Consider first the Reeb chord generators. Recall that the Reeb chords of $\Lambda(F,G)$ are in natural one to one correspondence with the Reeb chords of $K$. If $c$ is a Reeb chord of $K$ then we write $\hat c$ for the corresponding Reeb chord of $\Lambda(F,G)$. 

We calculate the grading of the chords of $\Lambda(F,G)$. In \cite[Lemma 3.4]{EES_noniso}, the grading of a Reeb chord was expressed in terms of the front projection as the sum of the Morse index of the critical point of the local function difference associated to the sheets of the front at the end and start points of the Reeb chord and the intersection number with the cusp edge, cooriented by minus the Reeb vector field, of a path in the Legendrian starting at the endpoint of the chord and ending at its start point. Using a path connecting the endpoints of $\hat c$ which lies in $K$ we find that the intersection index with the cusp edge is the same as that of $c$. The Morse index for $\hat c$ is one higher than that of $c$ and hence    
\[ 
|\hat c|=|c|+1,
\]
for any Reeb chord $c$ of $K$.

We now turn to the computation of the differential on $\Aa_{\Lambda(F,G)}$. Let $d$ denote the differential on $\Aa_{K}$ and $\hat d$ the differential on $\Aa_{\Lambda(F,G)}$. Let $\Phi_{F}\colon \Aa_{K}\to\Z$ and $\Phi_{G}\colon \Aa_{K}\to\Z$ be the augmentations induced by the Lagrangian fillings $F$ and $G$, respectively, then we have the following. 

\begin{lma}\label{l:dga}
For any Reeb chord $\hat c$ of $\Lambda(F,G)$,
\begin{equation}\label{eq:dhat}
\hat d(\hat c) = \Omega_{\Phi_{F};\Phi_{G}}(dc)-\Phi_F(c)+\Phi_G(c),
\end{equation}
where $\Omega_{\Phi_{F};\Phi_{G}}(1)=0$ and for a monomial $c_1c_2\dots c_m$, $m\ge 1$,
\begin{align*} 
&\Omega_{\Phi_{F};\Phi_{G}}(c_1c_2\dots c_m)=\\
&\sum_{j=1}^{m} 
(-1)^{|c_1|+\dots+|c_{j-1}|} \Phi_{F}(c_1)\dots\Phi_{F}(c_{j-1})\,\hat c_{j}\,\Phi_{G}(c_{j+1})\dots\Phi_{G}(c_m).
\end{align*}

It follows in particular that if for some Reeb chord $c$ of $K$, $dc=0$ and $\Phi_{F}(c)-\Phi_{G}(c)=\pm 1$ then $\Aa_{\Lambda(F,G)}$ is trivial, whereas if $F=G$ then $\Aa_{\Lambda(F,G)}=\Aa_{\Lambda(F,F)}$ admits the augmentation which takes all Reeb chord generators to zero and is hence non-trivial.
\end{lma}

\begin{proof}
We use rigid Morse flow trees on $\Lambda=\Lambda(F,G)$ to compute the differential on $\Aa_{\Lambda}$, see \cite{E}. Before going into the details (which were mostly worked out in \cite{EK}), we describe the structure of the computation that has three main steps. 

First, we degenerate $\Lambda$ and subdivide it into three pieces. Two of the pieces are essentially the Morse cobordisms $F^{\rm Mo}$ and $G^{\rm Mo}$ and the third piece is a product of the original knot $K$ with an interval and has Bott-degenerate Reeb chords. 

Second, we describe the flow trees of the three pieces. For the Morse cobordism pieces, this description is immediate once we know the rigid flow trees in the original $F^{\Mo}$ and $G^{\Mo}$. (As mentioned above, the rigid trees of $F^{\Mo}$ and $G^{\Mo}$ are used to compute the chain maps $\Phi_{F}$ and $\Phi_{G}$, see \cite[Lemmas 5.2, 5.11, and 5.12]{EHK} for details.) For the third piece we use a certain abstract perturbation scheme for flow trees. 

Third, we show that the differential computed via this abstract perturbation scheme gives a DGA that is stable tame isomorphic to the DGA calculated from usual flow trees, see Figure \ref{fig:morsebott}. We now turn to the actual argument.      

For small $\epsilon>0$, let
\[ 
\Lambda_{-}=\Lambda\cap\{x_2\le-\epsilon\},\quad \Lambda_0=\Lambda\cap\{-\epsilon< x_2<\epsilon\},\quad
\Lambda_{+}=\Lambda\cap\{x_2\ge\epsilon\}.
\] 
Then the front of $\Lambda_{0}$ is the front of $K$, lying over the $x_1$-axis, multiplied by the interval $(-\epsilon,\epsilon)$ in the $x_2$-axis and scaled by a function $h(x_2)$, $x_2\in(-\epsilon,\epsilon)$. If $\epsilon>0$ is small enough then the front of $\Lambda$ is cylindrical and scaled by $h(x_2)$ over a larger interval $x_{2}\in (-\delta,\delta)$. Here $h$ is a Morse function with a maximum at $0$, increasing on $(-\delta,0)$, and decreasing on $(0,\delta)$, see Section \ref{s:morsedim2}. 

Consider now a deformation of the front which makes the scaling function $h$ constant on $[-\epsilon,\epsilon]$, increasing on $(-\delta,-\epsilon)$, decreasing on $(\epsilon,\delta)$, and which fixes the front in $|x_2|\ge\delta$. After this deformation $\Lambda_-$ and $\Lambda_+$ are versions of $F^{\rm Mo}$ and $G^{\Mo}$, respectively, where the original Morse scaling function has been replaced by a monotone scaling function on the half interval with a degenerate rather than a Morse maximum at the endpoint. 

Let $\Lambda^{t}$, $0\le t\le 1$ denote the 1-parameter family of Legendrian submanifolds corresponding to this family of fronts. So that $\Lambda^{0}=\Lambda$ and $\Lambda^{1}$ is the Legendrian corresponding to the degenerate front. We will write $\Lambda^{t}_{\pm}$ and $\Lambda^{t}_{0}$ for the pieces of $\Lambda^{t}$ in the subdivision directly analogous to that of $\Lambda$ above. Note that $\Lambda^{1}$ has a closed interval of Reeb chords in $\Lambda^{1}_{0}$ for each Reeb chord of $K$. We consider the scaling Morse function $h$ as a function on these intervals. 

Let $\Gamma^{t}$ be a family of flow trees of $\Lambda^{t}$. We describe the limit $\Gamma^{1}$ of $\Gamma^{t}$ as $t\to 1$. The limit $\Gamma^{1}\cap\Lambda^{1}_{0}$ is a generalized flow tree in the sense of \cite[Section 3.3]{EK}. More precisely, since the vector fields associated to local function differences of the front has well defined limits as $t\to 1$ that have vanishing $x_2$-component, and $x_1$-component according to the local function differences of $K$ in $\Lambda_0$, the limit $\Gamma^{1}$ is a several level flow tree of the following form. 

It starts as a \emph{connector} which is a flow segment of $h$ in the Reeb chord manifold of the positive puncture (corresponding to the maximum of $h$ in the corresponding Bott interval of Reeb chords) of the flow tree. At the endpoint of the connector a flow tree in the slice with $x_2$-coordinate equal to that of the connector endpoint is attached at its positive puncture. At each negative puncture of the slice tree, a connector is attached, at the endpoints of the connectors slice flow trees are attached, etc until the connectors reach the slices $x_2=\pm\epsilon$. In particular, it follows that any generalized flow tree $\Gamma^{1}\cap \Lambda^{1}_{0}$ that arises in the limit ends in  the slices $x_{2}=\pm\epsilon$ at negative punctures which are the endpoints of a finite number of connectors. Finally, at the end of each connector in the slice $x_2=\pm\epsilon$, the positive puncture of a flow tree of $\Lambda_{\pm}^{1}$ is attached. 

In the notation of \cite{EK}, the above convergence result for flow trees $\Gamma^{t}\to\Gamma^{1}$ corresponds to ``convergence on the rough scale", and the required compactness result follows from a straightforward ODE-argument. In particular, for $\Gamma^{1}\cap\Lambda_0$ this is a simpler version of \cite[Lemmas 4.3 and 4.4]{EK}. The degeneration in \cite[Lemmas 4.3 and 4.4]{EK} corresponding to $\Gamma^{t}\to\Gamma^{1}$ here, is more elaborate and includes a sequence of four additional small perturbations near the Reeb chord manifolds that however do not affect the compactness argument.

Since we take $F^{\rm Mo}$ and $G^{\rm Mo}$ generic there are no flow trees of  $\Lambda_\pm^{1}$ (that are not contained in the slice $x_2=\pm\epsilon$) of formal negative dimension and all trees of formal dimension 0 are transversely cut out. If the same flow tree transversality (i.e., no trees of formal negative dimension and all trees of dimension $0$ transversely cut out) did hold for the generalized trees in $\Lambda_0^{1}$, then the description of rigid flow trees of $\Lambda$ would be straightforward. 

In this case, the rigid trees of $\Lambda$ would be in natural one to one correspondence with rigid generalized flow trees in $\Lambda_0^{1}$ capped off by a rigid tree in $\Lambda_{\pm}^{1}$ at each connector that ends at $x_2=\pm\epsilon$. To see this, we first use the compactness result discussed above, to conclude that the limit $\Gamma^{1}$ in $\Lambda^{1}$ of any sequence $\Gamma^{t}$ of rigid flow trees in $\Lambda^{t}$ is a generalized rigid flow tree in $\Lambda_{0}^{1}$ with connectors ending in the slices $x_{2}=\pm\epsilon$ where positive punctures of rigid trees in $\Lambda_{\pm}^{1}$ are attached. Second, we use a straightforward Morse-theory gluing argument to join the endpoints of the connectors to the flow lines of the positive punctures of rigid trees in $\Lambda^{1}_{0}$ to unique actual rigid trees of $\Lambda^{t}$ for small $t>0$. 

We give a short sketch of the Morse-theory gluing argument using the following notation from \cite{E}: A partial flow tree is a flow tree that is allowed to have special punctures, which are endpoints of flow line segments that are not necessarily critical points. For example if $\Gamma$ is a flow tree and we cut one of its edges then the two pieces $\Gamma'$ and $\Gamma''$ are partial flow trees with special punctures at the cut.  Consider now a configuration of a generalized rigid tree $\Gamma^{1}\cap\Lambda^{1}_{0}$ capped off by rigid trees at its connectors in $x_{2}=\pm\epsilon$ and consider partial flow trees on $\Lambda^{t}$ for $t>0$ small near the generalized flow tree and near the rigid trees with special negative punctures. The rigidity of the trees in the limit then implies the following. If the critical point of the local function difference in the slice $x_{2}=\pm\epsilon$ corresponding to the connector is a minimum then the negative special puncture of the partial flow trees near the generalized flow tree in $\Lambda_0^{1}$ is fixed, while the evaluation map at the positive special puncture of the flow trees corresponding to the rigid tree in $\Lambda_{\pm}^{1}$ is transverse to that fixed special negative puncture. If the critical point is a maximum it is vice versa, the positive special puncture is fixed and the evaluation map at the negative special puncture is transverse to it. This then implies that there is exactly one rigid flow tree of $\Lambda^{t}$ for small $t>0$ corresponding to the limit configuration. (For similar arguments in more involved situations, see \cite[Proposition 3.14 and Remark 3.16]{E}.)         


Returning to the situation under study, it is however not typically the case that flow tree transversality in $\Lambda_0^{1}$ holds for the following reason: if $\Delta$ is a flow tree of $K$ with positive puncture at $a$ and negative punctures at $b_1,\dots,b_m$ and of formal dimension $d=|a|-\sum_{j=1}^{m}|b_j|-1$ then the corresponding tree in the slice $x_2=0$, where $h$ has its unique local maximum, has positive puncture at $\hat a$ and negative punctures at $\hat b_1,\dots,\hat b_m$ and consequently has formal dimension 
\[ 
|\hat a| - \sum_{j=1}^{m}|\hat b_j| -1 = d+1-m.
\]
This is negative as soon as $d+1<m$. As shown in \cite{E}, it is possible to achieve flow tree transversality for flow trees by perturbing the Legendrian $\Lambda^{1}$ slightly, but, as the above shows, there will then generally be contributions to the DGA differential coming from higher dimensional moduli spaces. (It is particularly difficult to control such contributions via geometric perturbations when many negative punctures of some tree map to the same Reeb chord.) 

Degenerate cylindrical Legendrian submanifolds of the form of $\Lambda_{0}^{1}$ were studied in \cite[Sections 4--5]{EK}, where it was shown that the relevant count of rigid trees on $\Lambda_0$ is indeed determined by the rigid trees of $K$. The proof uses so called abstract flow trees and the results we use are \cite[Corollary 5.6 and 5.7]{EK}. Here Corollary 5.6 describes the abstract rigid flow trees in $\Lambda_0^{1}$ and together with the above decomposition of rigid flow trees of $\Lambda^{1}$ then directly implies that \eqref{eq:dhat} gives $\hat d$ as calculated by abstract flow trees (i.e., the second step mentioned in the beginning of the proof). Then Corollary 5.7 shows that the resulting DGA is stable tame isomorphic to the DGA calculated by actual rigid flow trees of $\Lambda^{1}$, geometrically perturbed to become generic for flow trees as in \cite{E} (i.e., the third step).         
 
For comparison with the higher dimensional case, see Remark \ref{r:M-Bholomorphic}, we give an overview of the argument that calculates the differential using the technique of abstract flow trees. The source of non-transversality for flow trees of $\Lambda^{1}$ is that the evaluation maps at the negative punctures of trees in slices are not transverse to the slice $x_2=0$ where the maximum of the function $h$ lies, see Figure \ref{fig:morsebott}. Abstract flow trees are perturbed out of this degenerate situation as follows. 

Flow trees in slices of $\Lambda^{1}_{0}$ lie in specific slices $\{x_2=\text{const}\}$. Abstractly perturbed flow trees are lifted flow trees in slices which have been perturbed slightly out of $\{x_2=\text{const}\}$ by a function on the boundary of the disk parameterizing the lifted flow tree (in a way compatible with flow tree breaking) with the following property. The function increases as we go along the boundary starting from the positive puncture and decreases back to the value at the positive puncture only after we passed the last negative puncture. As a consequence, for any abstractly perturbed generalized flow tree, the values of the evaluation maps at punctures are ordered in the $x_{2}$-direction along the Bott-manifolds according to their order along the boundary of the disk, starting from the positive puncture. This allows us to prove \eqref{eq:dhat} as follows (see Figure \ref{fig:morsebott}).     

To compute the differential $\hat d$ we count abstract rigid flow trees. The only way to rigidify an abstractly perturbed generalized flow tree is that one (and only one because of the ordering condition) of its negative punctures gets stuck at a Reeb chord in the middle. (Note that this is compatible with the dimension formula: if more than one negative puncture in a rigid tree gets stuck in the middle then its dimension is negative and if no puncture gets stuck then its dimension is $1$.) By the choice of perturbations, punctures before the one that gets stuck (in the order induced by the orientation of the boundary of the disks) fall off to the left and must continue as a tree in the Morse cobordism to the left and those after go analogously to the right, see Figure \ref{fig:morsebott}. The disks without negative punctures arise similarly when the initial flow tree is constant. 

Consider finally the two algebraic statements at the end of the lemma. Note that if $\hat c$ satisfies $\hat d\hat c=1$, then multiplication by $\hat c$ gives a chain homotopy $K\colon\Aa_{\Lambda}\to\Aa_{\Lambda}$ connecting the identity to the zero map. Also, since $\Aa_{\Lambda(F,F)}$ admits the zero augmentation, the map which acts as the identity in the coefficient ring and takes all generators $\hat c$ to zero is a non-trivial chain map to the ground ring with the trivial differential and hence the DGA is non-trivial.
\end{proof}

\begin{figure}[ht]
\centering
\includegraphics[width=.8\linewidth]{morsebott}
\caption{A perturbed flow tree configuration which contributes $\Phi_{F}(b_1)\hat b_{2}\Phi_{G}(b_{3})$ to $\hat d (\hat a)$. We start from the positive puncture $\hat a$ with a Morse flow line from the maximum in the Bott manifold of the Reeb chord $a$ of $K$. At the last point of this flow line a flow tree of $K$ grows out. This flow tree lies almost in a slice, the evaluation map at the first negative puncture maps to the left of the maximum in the Bott manifold of $b_1$, a flow line in the Bott manifold is attached and the flow tree eventually continues as a flow tree of the Morse cobordism corresponding to $F$, at the second negative puncture the evaluation map hits the maximum in the Bott manifold of $b_2$ and the corresponding negative puncture lies at $\hat b_2$, and the evaluation map at the third negative puncture lands to the right of the maximum in the Bott manifold of $b_3$ where a flow line which eventually continues as a flow tree of $G$ is attached. }
\label{fig:morsebott}
\end{figure}




\subsection{Morse cobordisms in higher dimension}\label{s:morsedim+}
The construction in Section \ref{s:morsedim2} carries over directly to higher dimensions.
If $\Sigma$ is a Legendrian $(n-1)$-submanifold in $\R^{2n-1}_{\st}$ that bounds an exact Lagrangian submanifold $W$ in the symplectization $\R\times \R^{2n-1}_{\st}$ then there is an associated Morse cobordism $W^{\Mo}$ in a half space of $\R^{2n}_{\st}$ with double points corresponding to the Reeb chords of $\Sigma$. Given two such Morse cobordisms in different half spaces $W^{\Mo+}$ and $V^{\Mo-}$ we can glue them along the boundary to get the projection of a Legendrian embedding $\Lambda(V,W)\subset\R^{2n+1}_{\st}$, which is then well defined up to translation along the $\R$-axis.

Furthermore, a filling $W$ induces a chain map $\Phi_{W}\colon\Aa_{\Sigma}\to\Z$ and Lemma \ref{l:dga} describing $\Aa_{\Lambda(V,W)}$ in terms of $\Phi_{V}$ and $\Phi_{W}$ holds. To show this one would need to develop the necessary Morse-Bott theory for holomorphic disks or extend the flow tree results of \cite{E,EK} to more complicated front singularities than cusp edges and swallow tails. The author believes that both these approaches work. Here we will however not develop them any further. 

Rather, to get examples that establish Theorem \ref{t:main} we will restrict attention to certain special Legendrian submanifolds and Lagrangian fillings constructed from the 2-dimensional case already considered and compute the DGA using a direct flow tree approach in these examples where all front singularities are as in the 2-dimensional case so that the flow tree results of \cite{E} can be applied directly. We describe these constructions next.

Let $K\subset\R^{3}_{\st}$ be a knot of Maslov index zero and consider its front. Consider also a Legendrian arc corresponding to half of the front of the unknot. The connect sum of the arc and $K$ gives a Legendrian arc $\gamma$ with front as shown in Figure \ref{fig:arc}.

\begin{figure}[ht]
\centering
\includegraphics[width=.7\linewidth]{arc}
\caption{The Legendrian arc $\gamma$.}
\label{fig:arc}
\end{figure}
 

\begin{lma}\label{lma:arcReeb}
The Legendrian arc $\gamma$ has two more Reeb chords than the knot $K$: to each Reeb chord of $K$ corresponds a Reeb chord of $\gamma$, furthermore there is one Reeb chord $b$ in the connect sum region and one central Reeb chord $a$ corresponding to the Reeb chord of the unknot. 
\end{lma}
  
\begin{proof}
See Figure \ref{fig:arc}.
\end{proof}

Assume that the knot $K$ has an exact Lagrangian filling $F$ in the symplectization $\R\times\R^{3}_{\st}$ with corresponding Morse cobordism $F^{\rm Mo}\subset \R^{4}_{\st}$. We construct a Lagrangian filling $\bar F$ with corresponding Morse cobordism ${\bar F}^{\rm Mo}$ for the Legendrian arc $\gamma$, see Figure \ref{fig:partcob}, as follows. 

\begin{figure}[ht]
\centering
\includegraphics[width=.6\linewidth]{partcob}
\caption{Schematic picture of the front of the Lagrangian filling $\bar F$. The solid curves not on the axes depict the caustic.}
\label{fig:partcob}
\end{figure}

Let $(x_0,y_0,x_1,y_1)$ be standard coordinates on $\R^{4}_{\st}$. Then the front of the Morse cobordism ${\bar F}^{\Mo}$ lies in $\R^{2}\times \R$ with coordinates $(x_0,x_1,z)$. We take it to start in the slice $\{x_0=0\}$ with the front of $\gamma$, then we remove the chord $b$ by a standard saddle cobordism at $x_0=\epsilon$ for $\epsilon>0$ small, see \cite[Section 6.5]{EHK}. This corresponds to region $\mathbf{I}$ in Figure \ref{fig:partcob}. It undoes the connected sum and results in a two component Legendrian 1-manifold, an arc $U$ which corresponds to half a Legendrian unknot and a copy of the Legendrian knot $K$. We next remove the kink in $U$ over a swallow tail in the front of the cobordism, corresponding to region $\mathbf{II}$ in Figure \ref{fig:partcob}. This results in an arc corresponding to one half of the standard unknot and a copy of $K$. Then fill $K$ with the Legendrian surface $F$ and fill the unknot arc by a quarter of the standard 2-dimensional unknot, with Morse cobordism that corresponds to rotating the front of $U$ an angle $\frac{\pi}{2}$. This corresponds to region $\mathbf{III}$ in Figure \ref{fig:partcob}. The resulting Lagrangian cobordism then has the topology of the boundary connect sum of $F$ and a disk, where the disk is a manifold with boundary with corners. 

Note next that we can define a DGA $\Aa_{\gamma}$ of the Legendrian arc $\gamma$ that is generated by Reeb chords and with differential $d$ that counts rigid flow trees just as for ordinary knots. Similarly if we have two Lagrangian fillings $\bar F$ and $\bar G$ of $\gamma$ as above then as for a closed knot $K$, $\bar F$ and $\bar G$ induce chain maps $\phi_{\bar F}, \phi_{\bar G}\colon \Aa_{\gamma}\to \Z$ by counting rigid flow trees on the corresponding Morse cobordisms. 

Furthermore, we can join the two fillings along $\gamma$ to create a Legendrian surface $\Lambda(\bar F,\bar G)$ with boundary and with the topology of the boundary connect sum $F\#_{\partial} G$ of $F$ and $G$. The Legendrian surface $\Lambda(\bar F,\bar G)$ has one Reeb chord $\hat c$ for each Reeb chord $c$ of $\gamma$, $|\hat c|=|c|+1$, and no other Reeb chords. As for $\gamma$ we can define a DGA  $\Aa_{\Lambda(\bar F,\bar G)}$ for $\Lambda(\bar F,\bar G)$ generated by Reeb chords and with differential $\hat d$ which counts rigid flow trees. The differential $\hat d$ can be described in terms of $d$ exactly as for the case of knots. More precisely, we have the following. 

\begin{lma}\label{l:halfcob}
The formula \eqref{eq:dhat} expresses $\hat d$ on $\Aa_{\Lambda(\bar F,\bar G)}$ in terms of $d$ on $\Aa_{\gamma}$. Furthermore, for the Reeb chord $b$ in the connect sum region, $\Phi_{\bar F}(b)=\Phi_{\bar G}(b)=1$.
\end{lma}   

\begin{proof}
Consider the front of $\Lambda(\bar F,\bar G)$ in $\R^{2}\times\R$ with coordinates $(x_0,x_1,z)$ as above. By construction it lies over the half plane $H=\{(x_0,x_1)\colon x_1\ge 0\}$. Note that there is a neighborhood $U$ of the boundary $\partial H$ over which the front of $\Lambda(\bar F,\bar G)$ agrees with that of the standard 2-disk Morse cobordism filling the Legendrian unknot. It follows that any rigid flow tree of $\Lambda(\bar F,\bar G)$ is disjoint from $U$. The proof of the first statement then follows from a word by word repetition of the proof of Lemma \ref{l:dga}. To see the last statement, note that both $\bar F$ and $\bar G$ start as saddle cobordisms with the obvious unique rigid flow trees emanating from $b$, see \cite[Section 6.5]{EHK}. 
\end{proof}

We next consider spinning constructions of Legendrian $n$-spheres and of the Legendrian $(n+1)$-submanifolds that correspond to their Lagrangian fillings. We start with the Legendrian $n$-spheres. 

Consider $S^{n-1}\subset\R^{n}$ and let $\xi$ be a coordinate on $S^{n-1}$. Let $(x(t),y(t),z(t))$, $t\in [0,\pi]$ be a parameterization of the Legendrian arc $\gamma$. Consider polar coordinates $(t,\xi)\in [0,\pi]\times S^{n-1}$ on $S^{n}$ and define $\Sigma_{K}$ as the Legendrian submanifold given by the embedding $\tau_K\colon S^{n}\to\R^{2n+1}_{\st}$,
\[ 
\tau_{K}(t,\xi)=\left(x(t)\xi, y(t)\xi,z(t)\right)\in \R^{2n}_{\st}\times\R = \R^{2n+1}_{\st}.
\] 
    
\begin{lma}\label{l:T_K}
The Reeb chords of $\Sigma_{K}$ are Bott-degenerate and are as follows.
\begin{itemize}
\item One central non-degenerate Reeb chord $a$ of grading $|a|=n$.
\item One $S^{n-1}$-family $S_{b}$ of Reeb chords corresponding to the connect sum Reeb chord $b$.
\item One $S^{n-1}$-family $S_{c}$ for each Reeb chord $c$ of $K$. 
\end{itemize}
\end{lma}

\begin{proof}
Immediate from the construction.
\end{proof}

Assume next that the knot $K$ bounds exact Lagrangian surfaces $F$ and $G$. As above, we let $\bar F$ and $\bar G$ denote the corresponding Lagrangian fillings of the Legendrian arc $\gamma$ and $\Lambda(\bar F,\bar G)$ the Legendrian embedding of $F\#_{\partial} G$ in $\R^{5}_{\st}$. Recall that the boundary of $F\#_{\partial} G$ is $S^{1}$ and let $S(F,G)$ denote the $(n+1)$-manifold 
\[ 
S(F,G)= (F\#_{\partial} G) \times S^{n-1} \ \cup_{S^{1}\times S^{n-1}} \ S^{1}\times D^{n}. 
\]
Note that if $F$ and $G$ are both 2-disks then $S(F,G)$ is an $(n+1)$-sphere.

Consider coordinates $(x_0,x_1,y_0,y_1,z)$ on $\R^{5}_{\st}$. By construction, the front of $\Lambda(\bar F,\bar G)$ lies in the half space $x_1\ge 0$ of $\R^{3}$ with coordinates $(x_0,x_1,z)$. Let $(x_0(\sigma),x_1(\sigma),y_{0}(\sigma),y_{1}(\sigma),z(\sigma))$, $\sigma\in F\#_{\partial} G$ be a parameterization of $\Lambda(\bar F,\bar G)$. Define $\Lambda^{n+1}(F,G)$ to be the image of the Legendrian embedding $T_{K}\colon S(F,G)\to \R^{2n+3}_{\st}$ given by  
\[ 
T_{K}(\sigma,\xi)=\left(x_0(\sigma),x_1(\sigma)\xi, y_0(\sigma),y_1(\sigma)\xi,z(\sigma)\right)\in \R^{2n+2}_{\st}\times\R = \R^{2n+3}_{\st},
\] 
where $\xi\in S^{n-1}$.

\begin{lma}\label{l:Bott-trees}
The Reeb chords of $\Lambda^{n+1}(F,G)$ agree with the Reeb chords of $\Sigma_K=\Lambda^{n+1}(F,G)\cap \{x_0=y_0=0\}$ with grading of the Reeb chord Bott manifolds shifted up by 1. The flow trees of $\Lambda^{n+1}(F,G)$ lie over half spaces
\[ 
H_{\xi} = \R\times R_{\xi}\subset \R\times\R^{n},
\]
where
\[  
R_{\xi}=\{r\cdot\xi\colon r\in[0,\infty), \xi\in S^{n-1} \}\subset\R^{n}.
\]
Furthermore, any flow tree in $H_{\xi}$ is a flow tree of the 2-dimensional Legendrian $\Lambda(\bar F,\bar G)$, and if $\Gamma$ is a flow tree in $H_{\xi}$ and if $Q\colon\R^{n}\to\R^{n}$ is a linear transformation in $SO(n)$ then $(1\times Q)(\Gamma)$ is a flow tree in $H_{Q\xi}$ (where $1\times Q\colon \R\times\R^{n}\to \R\times\R^{n}$). 
\end{lma}

\begin{proof}
This follows from the construction of $\Lambda^{n+1}(F,G)$ as an $SO(n)$ invariant Legendrian submanifold.
\end{proof}


\begin{rmk}\label{r:joincobordism}
Consider the Lagrangian projection $L^{n+1}(F,G)$ of $\Lambda^{n+1}(F,G)$ to $\R^{2n+2}_{\st}$ with coordinates $(x_0,y_0,\dots,x_n,y_n)$. Let 
\[ 
W_F^{\Mo}=L^{n+1}(F,G)\cap\{x_0 \le 0\}\quad \text{ and } \quad W_G^{\Mo}=L^{n+1}(F,G)\cap\{x_0\ge 0\}.
\]
Then $W_{F}^{\Mo}$ and $W_{G}^{\Mo}$ are Morse cobordisms corresponding to Lagrangian fillings $W_F$ and $W_G$ of the Legendrian sphere $\Sigma_{K}$. Thus we can also view $\Lambda^{n+1}(F,G)$ as the Legendrian lift of the exact Lagrangian immersion that result from joining the Morse cobordisms $W_F^{\Mo}$ and $W_G^{\Mo}$ along their common boundary, or using the notation above: 
\[ 
\Lambda^{n+1}(F,G)=\Lambda(W_F,W_G).
\]
\end{rmk}


Write $\Lambda=\Lambda^{n+1}(F,G)=\Lambda(W_F,W_G)$. We next study properties of the DGA $\Aa_{\Lambda}$. Consider first the Reeb chord generators. Perturbing the Legendrian by small Morse functions with two critical points on the $S^{n-1}$ Bott manifolds we find the following if $c_1,\dots,c_m$ denotes the Reeb chords of $K$ (see Lemma \ref{l:T_K}).
\begin{itemize}
\item Reeb chords $\hat c_1,\dots,\hat c_m$, where $|\hat c_j|=|c_j|+1$, $j=1,\dots,m$.
\item Reeb chords $\tilde c_1,\dots,\tilde c_m$, where $|\tilde c_j|=|c_j|+n$.
\item The connect sum Reeb chords $\hat b$ with $|\hat b|=1$ and $\tilde b$ with $|\tilde b|=n$ 
\item The central Reeb chord $a$ with $|a|=n$.
\end{itemize}

Let ${\hat{\Aa}}_{\Lambda}$ denote the unital subalgebra of $\Aa_{\Lambda}$ generated by $\hat c_1,\dots,\hat c_m$ and $\hat b$ and let $\tilde{\mathcal{I}}_{\Lambda}$ denote the ideal generated by $a,\tilde b,\tilde c_1,\dots,\tilde c_m$. Writing $D$ for the differential on $\Aa_{\Lambda}$, we then have the following result.

\begin{lma}\label{l:augandDGAhigh}
Both the subalgebra $\hat{\Aa}_{\Lambda}$ and the ideal $\tilde{\mathcal{I}}_{\Lambda}$ 
are invariant under $D$. Furthermore, for any generator $\hat c$ of $\hat{\Aa}_{\Lambda}$
\[ 
D\hat c = \hat d \hat c,
\]
where the differential $\hat d$ is as in Lemma \ref{l:halfcob}.

It follows in particular that if some Reeb chord $c$ of the Legendrian knot $K$ in $\R^{3}_{\st}$, used to construct $\Lambda=\Lambda^{n+1}(F,G)$, satisfies $dc=0$ and $\Phi_{F}(c)-\Phi_{G}(c)=\pm 1$ then $\Aa_{\Lambda}$ is trivial, whereas if $F=G$ then $\Aa_{\Lambda}$ admits the zero augmentation and is hence non-trivial.
\end{lma}

\begin{proof}
The Legendrian submanifold $\Lambda$ is a small perturbation of a Legendrian submanifold with $S^{n-1}$ Bott manifolds of Reeb chords and the generators $\hat c_j$ and $\hat b$ correspond to the minima in their respective Bott manifolds. Furthermore, by construction, all singularities of the front of $\Lambda$ are as in the 2-dimensional case, i.e.~there are cusp-edges of codimension one and swallow-tails of codimension two, and no other front singularities. Therefore the flow tree result of \cite{E} can be applied directly to compute the differential $D$. 

It is straightforward to check that the rigid flow trees with positive puncture at $\hat c_j$ or $\hat b$ after perturbation are in natural one to one correspondence with the following configurations on the Bott-degenerate Legendrian submanifold. A flow tree over the half space $H_\xi$, see Lemma \ref{l:Bott-trees}, where $\xi$ is the coordinate of the minimum in the Bott manifold of the positive puncture, with flow lines emanating from the negative punctures of the tree and ending at the minima of the corresponding Bott manifolds. It follows that $\hat{\Aa}_{\Lambda}$ is invariant under $D$. The differential is as claimed since the rigid flow trees in $H_\xi$ for any $\xi$ gives $\hat d$.

To see the second statement we show that any flow tree with positive puncture at a chord $\tilde c_j$, $\tilde b$, or $a$ and with negative punctures only at chords $\hat c_j$ and $\hat b$ has dimension at least $n-1>0$. Consider such a flow tree and note that in the Bott-degenerate limit it corresponds to a
several level flow tree of the following form.

It starts as a connector (i.e., a flow line) in the Bott manifold of the positive puncture emanating from the maximum. At the end point of this connector a flow tree in a half space $H_{\xi}$ is attached. At the negative punctures of this flow tree connectors in Bott manifolds are attached. These connectors either end at critical points or at some other point where the positive puncture of a flow tree in a half space $H_{\eta}$ is attached. The configuration continues like this with further levels until all connectors reach critical points which then corresponds to the neagtive punctures of the original tree. 

Since the dimension of any flow tree in a half space is non-negative it follows from the grading formulas for the generators of $\hat{\Aa}_{\Lambda}$ that the dimension of the flow tree considered is at least $n-1>0$. Thus any rigid flow tree with a positive puncture at a generator of $\tilde{\mathcal{I}}_{\Lambda}$ must have at least one negative puncture at a generator of $\tilde{\mathcal{I}}_{\Lambda}$.

The last statement follows exactly as the corresponding statement in Lemma \ref{l:dga}.       
\end{proof}


\begin{rmk}\label{r:M-Bholomorphic}
A straightforward extension of the abstract flow tree techniques from \cite{EK}, used in Lemma \ref{l:dga} to compute the differential, from 1-dimensional to higher dimensional Bott manifolds, gives the full differential $D$ on $\Aa_{\Lambda}$ as follows. For generators of $\hat{\Aa}_{\Lambda}$ the differential is given by $\hat d$ as stated in Lemma \ref{l:augandDGAhigh} and for other generators we have 
\begin{align*}
D a &= \tilde b,\\
D \tilde b &=0,\\
D\tilde c &= \tilde{\Gamma}(\hat d \hat c),
\end{align*}
where $\tilde{\Gamma}$ acting on a monomial $\hat c_{i_1}\dots\hat c_{i_r}$ satisfies
\begin{align*} 
\tilde{\Gamma}(\hat c_{i_1}\dots\hat c_{i_r}) &=
\tilde c_{i_1}\dots\hat c_{i_r}\\
&+
(-1)^{|\hat c_{i_1}|}\hat c_{i_1}\tilde c_{i_{2}}\dots\hat c_{i_r}
\dots + (-1)^{|\hat c_{i_1}|+\dots + |\hat c_{i_{r-1}}|}\hat c_{i_1}\dots\hat c_{i_{r-1}}\tilde c_{i_{r}}.
\end{align*}
In fact this formula simplifies further if we use abstract flow trees to compute the differential on $\Lambda(\bar F,\bar G)$. In this case $\hat d c_{i_{0}}=\sum_j \hat c_{i_j}$ is a linear combination of other generators, and $D\tilde c_{i_0} =\sum_j\tilde c_{i_j}$ is the linear combination of the corresponding generators.
\end{rmk}


\section{Non-loose Legendrian spheres}\label{S:nloose}
In this section we first describe a specific Legendrian knot $K$ in $\R^{3}_{\st}$ which has two Lagrangian disk fillings that induce distinct augmentations. We then prove that the associated Legendrian $n$-spheres, $n\ge 2$, are in fact not loose. It should be mentioned that the knot $K$ is by no means unique with the property of supporting disk fillings with different augmentations. Other Legendrian knots with this property similarly lead to examples of non-loose Legendrian $n$-spheres with trivial DGA. How the Legendrian isotopy class of the resulting Legendrian spheres depends on the knots and the fillings is presently unknown.

\subsection{A Legendrian knot with two disk fillings}\label{s:twodisks}
Consider the the Legendrian knot $K$ in Figure \ref{fig:exknot}. This knot belongs to the isotopy class $\overline{9_{46}}$ and its Lagrangian disk fillings are described in \cite{BC,CNS}. 

\begin{figure}[ht]
\begin{subfigure}{.4\textwidth}
\centering
\includegraphics[width=.9\linewidth]{frontexknot}
\end{subfigure}%
\begin{subfigure}{.6\textwidth}
\centering
\includegraphics[width=.9\linewidth]{lagexknot}
\end{subfigure}
\caption{Front and Lagrangian projections of the knot $K$.}
\label{fig:exknot}
\end{figure}
 
Let $c_0$ and $c_1$ be the Reeb chords of $K$ as shown in Figure \ref{fig:exknot}. We use the notion of \emph{saddle cobordism} from \cite{EHK}: this is an elementary cobordism which topologically corresponds to attaching a $1$-handle and geometrically corresponds to shrinking and removing a Reeb chord. In \cite[Section 6.5]{EHK} the cobordism map of a special type of saddle cobordisms were calculated. This class of saddle cobordisms were called \emph{simple} and are characterized by the following property. Let $c$ be the Reeb chord of $K$ that shrinks, then any holomorphic disk with boundary on $K$, with one positive puncture at some Reeb chord $b\ne c$, $k\ge 2$ positive punctures at $c$, and any number of negative punctures have formal dimension $>k$. 

\begin{lma}\label{l:simplecob}
There exists two simple Lagrangian saddle cobordisms $C_0$ and $C_1$ with positive end at the knot $K$ and negative end at a 2-component Legendrian unlink. Furthermore, with notation as in Figure \ref{fig:exknot}, the Reeb chords $c_j$ of $K$ satisfy $|c_j|=0$, $j=0,1$, $dc_j=0$, and the chain maps of the cobordisms satisfy
\[ 
\Phi_{C_{i}}(c_j)=
\begin{cases}
\pm 1 &\text{ if }i=j,\\
c_j &\text{ if }i\ne j.
\end{cases}
\]
\end{lma}

\begin{proof}
It is straightforward to check that $c_0$ and $c_1$ are both contractible. In particular they can be chosen shorter than all other Reeb chords and hence by Stokes theorem $dc_j=0$, $j=0,1$.

To rule out disks with more than two positive punctures at $c_j$ and only one other positive puncture we note that the boundary of such a disk must come in towards the puncture along the lower branch at $c_j$. Following the boundary of such a disk backwards from $c_j$, we see that it must turn at a positive puncture (otherwise the disk would go out into the exterior of the knot digram which would contradict the maximum principle). Thus any disk with more than two positive punctures at $c_j$ has more than two other positive punctures and the cobordism is simple. It then follows from the proof of \cite[Proposition 6.17]{EHK} that the chain map for the elementary saddle cobordism satisfies the equation as claimed.

To see that the negative ends of the cobordisms are as stated we refer to Figure \ref{fig:pinchexknot} 
\end{proof}

\begin{figure}[ht]
\centering
\includegraphics[width=.5\linewidth]{pinchexknot}
\caption{The negative end of the simple saddle cobordism that shrinks and removes $c_0$.}
\label{fig:pinchexknot}
\end{figure}


Let $C_{j}$ be one of the cobordisms in Lemma \ref{l:simplecob}. Pick a Legendrian isotopy from the unlink at the negative end of $C_{j}$ to the standard unlink on two components. As explained in \cite{EHK} this isotopy gives an exact Lagrangian cylinder that gives a chain map to the algebra of the unlink. Then cap the components of the unlink by two standard Lagrangian disks that in turn give trivial chain maps to the ground ring $\Z$. Writing $F_{j}$ for the Lagrangian disk constructed by joining these cobordisms we have the following:
\begin{lma}\label{l:diskfilling}
With Reeb chord notation as above, the chain maps $\Phi_{F_{j}}$, $j=0,1$ satisfy
\begin{equation*}
\Phi_{F_{i}}(c_j)=
\begin{cases}
\pm 1 &\text{ if }i=j,\\
0 &\text{ if }i\ne j.
\end{cases}
\end{equation*}
\end{lma}
\begin{proof}
For $i=j$ this is a consequence of Lemma \ref{l:simplecob}. To see that $\Phi_{F_i}(c_j)=0$ if $i\ne j$, we note that $c_j$ is a mixed chord of the two component unlink (i.e.~its endpoints belong to different components) at the negative end of $C_i$ and that $F_i-C_i$ is a two component Lagrangian filling with disk components. Hence the moduli of disks with positive puncture at $c_j$ and boundary on $F_i-C_i$ is empty for topological reasons. It follows that $\Phi_{F_i}(c_j)=\Phi_{F_i-C_i}(c_j)=0$.
\end{proof}

Furthermore, the linearized contact homology of $K$ with respect to the augmentation induced by $C_{j}$ is zero except in degree 1 where it has rank 1, reflecting the topology of the disk. See \cite[Section 7 and Theorem 7.2]{EHK} for the definition of and necessary result on linearized contact homology. 



\subsection{Lifts of Hamiltonian isotopies and symplectic neighborhoods}\label{s:lifts}
In this section we include a short discussion on lifts of symplectic data in $\R^{2n}_{\st}$ to contact data in $\R^{2n+1}_{\st}$. The material presented is certainly well known. We include it to prepare for the proof of Theorem \ref{t:main}. 

\subsubsection{Contact lifts of Hamiltonian isotopies}\label{ss:ham}
We start by describing the lift of a Hamiltonian isotopy to a contact isotopy. Consider an exact symplectic manifold $P$ with symplectic form $\omega=-d\theta$ and the associated contact manifold $P\times \R$ with contact from $dz-\theta$. Our main example is $\R^{2n}_{\st}$ with $\theta=y\cdot dx$ and $\R^{2n+1}_{\st}$, but we will use the construction in more general situations as well.

Let $H\colon P\to\R$ be a Hamiltonian function. The \emph{Hamiltonian vector field} of $H$ is the vector field defined by the equation:
\[ 
dH = \iota_{X_{H}}\omega,
\]  
where $\iota_{Z}$ denotes contraction with the vector field $Z$. We find in particular that
\[ 
L_{X_{H}}\omega = d(\iota_{X_{H}}\omega)+\iota_{X_{H}}d\omega=0,
\]
where $L_{Z}$ denotes the Lie derivative along the vector field $Z$, and it follows that the flow of $X_{H}$ preserves $\omega$.

Consider now the lift $V$ of the vector field $X_{H}$ to $P\times\R$ given by
\[ 
V=X_{H} + (\theta(X_{H})-H)\partial_{z},
\]
where we write $X_{H}$ for the vector field on $P\times\R$ that maps to $X_{H}$ in $P$ and has $\partial_{z}$-component equal to $0$. Then with $\alpha=dz-\theta$:
\begin{align*}
L_{V}\alpha &= d(\iota_{V}\alpha) + \iota_{V}d\alpha\\ 
&= d(\theta(X_{H})-H-\theta(X_{H}))+\iota_{X_{H}}\omega =0.
\end{align*}
Thus the flow of $V$ preserves the contact form. We call $V$ the \emph{contact lift} of the Hamiltonian vector field $X_{H}$.

\subsubsection{Contact lifts of symplectic neighborhoods of Lagrangian submanifolds}\label{ss:lag}
Second, we consider contact lifts of a standard neighborhood of a compact and connected Lagrangian submanifold $L\subset \R^{2n}$. We will apply this result in the case when $L$ is a disk. Moser's lemma implies that there is a neighborhood of $L$ in $\R^{2n}$ that is symplectomorphic to a neighborhood of the $0$-section in the cotangent bundle $T^{\ast}L$ of $L$ with its standard symplectic form $-d(p\cdot dq)$. In other words there is a map
$\psi\colon D^{\ast}L\to \R^{2n}$, defined in a disk bundle neighborhood of the 0-section and agreeing with the inclusion along the $0$-section, such that $\psi^{\ast}(d(y\cdot dx))=d(p\cdot dq)$. It follows that the $1$-form $\psi^{\ast}(y\cdot dx)-p\cdot dq$ is closed and vanishes along the $0$-section. By homotopy invariance of de Rham cohomology it then follows that there exists a function $h\colon D^{\ast}L\to\R$ such that $dh=\psi^{\ast}(y\cdot dx)-p\cdot dq$.  Consider the map $\Psi\colon (D^{\ast}L)\times\R \to\R^{2n+1}_{\st}$,
\begin{equation}\label{eq:contactext} 
\Psi(\xi,\zeta)=(\psi(\xi),\zeta+h(\xi))\in\R^{2n}\times\R.
\end{equation}
Then
\[ 
\Psi^{\ast}(dz-y\cdot dx)=(d\zeta +dh) -(p\cdot dq +dh)= d\zeta -p\cdot dq, 
\]
so that $\Psi$ intertwines the standard contact form on $\R^{2n+1}_{\st}$ with the natural 1-jet contact form on $(D^{\ast}L)\times\R$. Assume next that $L$ is the projection of a Legendrian submanifold $\tilde L\subset \R^{2n+1}_{\st}$. Then the $z$-coordinate along $\tilde L$ is a function on $L$ which satisfies $dz=y\cdot dx$. In particular since $p=0$ along the $0$-section in $D^{\ast}L$ we find that $z=h+C$, where $C$ is some constant, along $L$.    

\subsubsection{Contact lifts of symplectic neighborhoods of isotropic arcs}\label{ss:arc}
Finally, we consider similar results starting from an arc $\gamma\subset \R^{2n}_{\st}$, rather than from a Lagrangian submanifold. Since $\gamma$ is 1-dimensional it is automatically isotropic. It has an isotropic lift to $\R^{2n+1}_{\st}$ as follows. If $\gamma(t)$, $t\in[0,\ell]$ is a parameterization of $\gamma$ then define the $z$-coordinate of its lift $\tilde{\gamma}$ at $t$ as
\begin{equation}\label{eq:zcoordandh} 
z(t)=\int_{\gamma|_{[0,t]}}y\cdot dx.
\end{equation}  
Consider next a small neighborhood $U$ of $\gamma$ of the form
\[ 
\psi\colon (-\epsilon,\ell+\epsilon)\times(-\epsilon,\epsilon)\times B^{2n-2}_{\epsilon}\to\R^{2n}_{\st},
\]
where $B^{2n-2}_{\epsilon}$ is the standard ball in $\R^{2n-2}_{\st}$ and where $\psi|_{[0,\ell]\times 0\times 0}$ is a parameterization of $\gamma$. Using Moser's trick we find that we can choose this neighborhood with coordinates $(t,s;u,v)\in\R^{2}\times\R^{2n-2}$ so that
\[ 
\psi^{\ast}(dx\wedge dy)= dt\wedge ds + du\wedge dv,
\]
where $dx\wedge dy$ is the standard symplectic form on $\R^{2n}$ and $du\wedge dv$ is the symplectic form on $\R^{2n-2}_{\st}$. As above, we then get a contact neighborhood $U\times\R$ of $\gamma\times\R$ parameterized by $(\psi,z)$ where 
\[  
z=\zeta+h(t,s;u,v).
\]
Here $\zeta$ is the $\R$-coordinate in $U\times\R$, with contact form $d\zeta-(sdt+v\cdot du)$, and $dh=\psi^{\ast}(y\cdot dx) - (sdt+v\cdot du)$. In particular, $h(t,0,0,0)=z(t)$, see \eqref{eq:zcoordandh}, and for $\epsilon>0$ small enough
\[ 
|h(t,s,u,v)-h(t,0,0,0)|=\mathcal{O}(\epsilon^{2}).
\] 
To see this note that the coordinate chart has bounded derivative, hence, using $dh=\psi^{\ast}(y\cdot dx) - (sdt+v\cdot du)$, we find
\[ 
|dh(t,s,u,v)-dh(t,0,0,0)|=\mathcal{O}(\epsilon),
\]
and the result follows.

\subsection{Proof of Theorem \ref{t:main}}
Let $K$ be the Legendrian knot in Figure \ref{fig:exknot} with exact disk fillings $F_{0}$ and $F_{1}$, see Lemma \ref{l:diskfilling}. For $n\ge 3$, consider the Legendrian $(n-1)$-spheres $\Sigma_{K}$, see Lemma \ref{l:T_K} with corresponding fillings $W_{i}=W_{F_i}$, $i=0,1$ and the Legendrian $n$-spheres $\Lambda(W_{i},W_{j})\subset \R^{2n+1}_{\st}$, see Remark \ref{r:joincobordism}. To unify notation we let $\Sigma_{K}=K$ and $W_{i}=F_{i}$ for $n=2$. It follows from Lemmas \ref{l:dga} and \ref{l:augandDGAhigh} that the DGA $\Aa_{\Lambda(W_{i},W_{j})}$ is trivial if and only if $i\ne j$. We next prove the following result which is more specific than Theorem \ref{t:main} and from which the latter then follows immediately. 

\begin{theorem}
The Legendrian $n$-sphere $\Lambda(W_{0},W_{1})$, $n\ge 2$ is not loose.
\end{theorem}

\begin{proof}
Write $\Lambda=\Lambda(W_0,W_1)$ and assume that it is loose. Let $\psi\colon (B_{0},\Delta_{0})\to(\R^{2n+1}_{\st},\Lambda)$ be a loose chart for $\Lambda$. Since $\psi(\Delta_{0})\subset\Lambda$ is an embedded disk there exists a point $\xi$ in one of the hemispheres of $\Lambda$ in the complement of $\psi(\Delta_0)$, and hence also in the complement of $\psi(B_0)$ by the definition of a loose chart. Assume for definiteness that $\xi\in W_{1}$ and consider the Reeb axis $\{\xi\}\times\R$ through it. We claim that we can contact isotope $\psi$ by an isotopy through loose charts that is fixed on $\Delta_0$ and ends with a loose chart that is disjoint from the Reeb axis $\{\xi\}\times\R$.   

To see this we note that the complement of $\psi(B_0)$ in $\R^{2n+1}_{\st}$ is connected and that, by compactness, the projection of $\psi(B_0)$ to $\R^{2n}_{\st}$ lies inside some finite ball $D$. Hence we can find an isotropic path $\gamma$ in $\R^{2n+1}_{\st}$ in the complement of $\psi(B_0)$ connecting $\xi$ to a point which projects outside the finite ball $D$. Furthermore, by general position this path can be chosen so that its projection into $\R^{2n}_{\st}$ is embedded and does not intersect the projection of $\Lambda$ except at the endpoint $\xi$. We can then find coordinates on $\R^{2n}_{\st}$ in a neighborhood $N$ of the projection of this path of the form 
\[ 
(t,s,u,v)\in (-\epsilon,\ell+\epsilon)\times(-\epsilon,\epsilon)\times B^{2n-2}_{\epsilon}, 
\]
where we use notation as in Section \ref{ss:arc}: $\epsilon>0$ is small, $B^{2n-2}_{\epsilon}$ is the standard symplectic $(2n-2)$-ball of radius $\epsilon$, and the symplectic form on $\R^{2n}_{\st}$ corresponds to $dt\wedge ds+du\wedge dv$. Furthermore, the isotropic path corresponds to $\{0\le t\le \ell;\,s=0;\,u=0;\,v=0\}$ and the Lagrangian around $\xi$ to $\{t=0;\,v=0\}$. As explained in Section \ref{ss:arc}, the lift of this coordinate chart to $\R^{2n+1}_{\st}$ is a coordinate chart $N\times\R\to\R^{2n}_{\st}\times\R$ with the coordinates along $\R^{2n}_{\st}$ as above and with the remaining coordinate given by  
\[ 
z=\zeta + h(t,s,u,v),
\]
where $h(t,0,0,0)$ is the $z$-coordinate along the isotropic path and where, provided the neighborhood is chosen small enough,
\begin{equation}\label{eq:arcchange}
|h(t,s,u,v)-h(t,0,0,0)|=\mathcal{O}(\epsilon^{2}).
\end{equation}
Consider the Hamiltonian $H$ supported in $V$ and given by
\[ 
H(t,s,u,v)= \kappa(s) f(t) \phi(u,v),
\]
where the factors have the following properties:
\begin{itemize}
\item $\phi(u,v)=1$, for $|(u,v)|\le\frac14\epsilon$,  $\phi(u,v)=0$, for $|(u,v)|\ge\frac34\epsilon$, and $|d\phi(u,v)|=\mathcal{O}(\epsilon^{-1})$,
\item $f(t)=1$ for $t\in [0,\ell]$, $f(t)=0$ for $t\notin [-\frac12\epsilon,l+\frac12\epsilon]$,  and $|f'(t)|=\mathcal{O}(\epsilon^{-1})$,
\item $\kappa(s)=\mathcal{O}(\epsilon)$, $\kappa'(s)=1$ for $|s|\le\frac14\epsilon$ and $\kappa(s)=\kappa'(s)=0$, $|s|>\frac34\epsilon$. 
\end{itemize}
It then follows that the derivative of the $z$-coordinate for the contact lift, see Section \ref{ss:ham}, satisfies
\begin{equation}\label{eq:zcoord} 
|(sdt+v\cdot du)(X_{H})-H|=\mathcal{O}(\epsilon).
\end{equation}
Note that the time $\ell$ flow of $X_H$ shifts $\xi$ to the endpoint of the projection of the isotropic path $\gamma$ and that the time $\ell$ contact lift $Z^{\ell}$ of the flow of $X_{H}$ pushes a small part of $\Lambda$ along $\gamma$. By the estimates of the $z$-coordinates of the coordinate change \eqref{eq:arcchange} and of the flow \eqref{eq:zcoord}, we find that $Z^{\ell}(\Lambda)$ lies in a small neighborhood of $\Lambda\cup\gamma$ and hence $\psi(B_0)$ is a loose chart of $Z^{\ell}(\Lambda)$, see Figure \ref{fig:isotropic}. It follows that $Z^{-\ell}(\psi(B_0))$ is a loose chart for $\Lambda$ and 
\[  
(\xi\times\R)\,\cap\, Z^{-\ell}(\psi(B_0))=\varnothing.
\]
For simplicity we change notation and write $\psi$ instead of $Z^{-\ell}\circ \psi$. 

\begin{figure}[ht]
\centering
\includegraphics[width=.5\linewidth]{isotropic}
\caption{Pushing a small part of $\Lambda$ around $\xi$ along an isotropic path.}
\label{fig:isotropic}
\end{figure}


The two disks of $\Lambda(W_0,W_1)$ lie in two half spaces of $\R^{2n+1}_{\st}$ separated by a hyperplane which we take to be $\{x_n=0\}$, with $W_0\subset\{x_n\le 0\}$ and $W_1\subset\{x_n\ge 0\}$, in standard coordinates $(x_1,y_1,\dots,x_n,y_n,z)$ on $\R^{2n+1}_{\st}$. Consider the Lagrangian disk $L$ which is the projection of the intersection of $\Lambda$ with the half space $\{x_n\ge \delta\}$ for small $\delta>0$. Since $\xi\in L$, we can parameterize a small neighborhood of $L$ by symplectic coordinates $(q,p)$ in a disk bundle neighborhood $D^{\ast}L$ of the $0$-section in the cotangent bundle of $L$, where the point $(0,0)$ corresponds to $\xi$, and extend this to a contact isomorphism, see Section \ref{ss:lag}.   

Consider the Hamiltonian on $D^{\ast}L$ given by
\[ 
H(q,p)= \sigma(|q|)\tau(|p|)q\cdot p,
\]
where, for small $\epsilon>0$, $\sigma=1$ outside an $2\epsilon$-neighborhood of the boundary $\partial L$ and $\sigma=0$ inside an $\epsilon$-neighborhood of $\partial L$, and where $\tau$ is a similar cut off function in the fiber directions, equal to $1$ near the center and $0$ near the boundary.
Consider the corresponding Hamiltonian vector field $X_H$, and let $V$ denote the corresponding contact lift (see Section \ref{ss:ham}): 
\[ 
V=X_{H}+ (\alpha(X_{H})-H)\pa_{z}.
\] 
The time $t$ flow $Z^{t}$ of $V$ is a contactomorphism, that leaves the Legendrian lift of $L$ invariant since $H$, $\theta(X_{H})$, and the $p$-component of $X_{H}$ all vanish along $L$. Furthermore, $Z^{t}$ fixes a neighborhood of the boundary of $D^{\ast}L\times\R$ and hence we can extend it by the identity to all of $\R^{2n+1}_{\st}$. Denote the corresponding flow of $\R^{2n+1}_{\st}$ by $Z^{t}$ still. 

Since the Hamiltonian flow along the disk $L$ pushes radially away from its center eventually spreading any little disk around the origin over the complement of the $2\epsilon$-neighborhood of the boundary, it is clear that there exists a (possibly large) $T>0$ such that the projection of $Z^{T}(\psi(B_0))$ to $\R^{2n}_{\st}$ is disjoint from the complement of a small neighborhood of the boundary of $L$. For example, by taking $T>0$ sufficiently large we can make $Z^{T}(\psi(B_0))$ disjoint from the intersection $L_1$ of $\Lambda$ with the half space $\{x_2\ge 2\delta\}$ provided $\epsilon\ll\delta$. Fix such a $T>0$. 

Note that $\psi'(B_0)=Z^{T}(\psi(B_0))$ is a loose chart for $\Lambda$. Let $U'\subset U$ be small nested tubular neighborhoods of $L_1$ such that the projection of $\psi'(B_0)$ is disjoint from $U$. Fix a Hamiltonian function $H\colon \R^{2n}_{\st}\to\R$ with the following properties.
\begin{itemize}
\item $H=0$ in $U'\cup \{x_{2}\le \delta\}$,
\item $H=-1$ in $\{x_2\ge 2\delta\}-U$,  
\end{itemize} 
see Figure \ref{fig:nbhd}.

\begin{figure}[ht]
\centering
\includegraphics[width=.6\linewidth]{nbhd}
\caption{The Hamiltonian equals $0$ near $L$ and in the left half plane, and equals $-1$ in the complement of $U$ in the right half plane.}
\label{fig:nbhd}
\end{figure}


Let $V$ denote the contact lift of the Hamiltonian vector field of $H$. Again the flow of $V$ leaves $\Lambda$ invariant. Furthermore, since the projection of $\psi'(B_0)$ is disjoint from $U$ the flow of $V$ shifts the intersection of $\psi'(B_0)$ with $\{x_2\ge 2\delta\}$ in the $z$-direction with unit speed. In particular, by flowing for sufficiently large time we find for any $M>0$ a loose chart $\psi_{M}(B_0)$ for $\Lambda$ whose intersection with $\{x_2\ge 2\delta\}$ lies in the region $\{z\ge M\}$. This however leads to a contradiction as follows.  

Cut $\Lambda$ along $\{x_2=2\delta\}$, remove $L_1\subset W_1$, and insert instead a corresponding piece $L_0\subset W_0$. The lift of the resulting exact Lagrangian sphere is then $\Lambda'=\Lambda(W_{0},W_{0})$. Since $L_0$ is compact its Legendrian lift is bounded in the $z$-direction and thus for $M>0$, $\psi_M(B_0)$ is a loose chart for $\Lambda'$ which contradicts Lemmas \ref{l:dga} and \ref{l:augandDGAhigh}.    
\end{proof}

 
\bibliographystyle{utcaps}
\bibliography{myrefs}



\end{document} 
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\end{thebibliography}\endgroup
