Finite Order Invariants for $(n,2)$-Torus Knots \\ and the Curve $Y^2 = X^3 + X^2$
Svetlana Tyurina${}^{*}$ \ {}\ and \ {} \ Alexander Varchenko${}^{**,1}$
We describe the algebra of finite order invariants on the set of all $(n,2)$-torus knots.
math0402185.tex
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\begin{document}






\title[Finite Order Invariants for $(n,2)$-Torus Knots]
{Finite Order Invariants for $(n,2)$-Torus Knots \\ and the Curve
$Y^2 = X^3 + X^2$}
\author[S. Tyurina and A. Varchenko]
{Svetlana Tyurina${}^{*}$ \ {}\ and \ {} \ Alexander Varchenko${}^{**,1}$ }
\thanks{${}^1$Supported in part by NSF grant  DMS-0244579}






\begin{abstract}
We describe the algebra of finite order invariants on the set of all
$(n,2)$-torus knots.

\end{abstract}






\maketitle

\centerline{\it ${}^{*}$OASIS,
University of North Carolina at Chapel Hill,}
\centerline{\it Chapel Hill, NC 27599, USA}
\centerline{svetlana@email.unc.edu}

\medskip
\centerline{\it ${}^{**}$Department of Mathematics,
University of North Carolina at Chapel Hill,}
\centerline{\it Chapel Hill, NC 27599-3250, USA}
\centerline{anv@email.unc.edu}


\bigskip





\centerline{February, 2004}

\bigskip

This paper is an extended exposition of the talk \cite{Ty} given by the
first author. The authors thank S. Duzhin and A. Sossinsky for interest to this
work and S. Chmutov for useful  discussions.

\bigskip

%\noindent
%{\bf 1. .}




Consider the $\Q$-algebra $V$ of Vassiliev finite order knot invariants, see
for example \cite{B, CDL}. The algebra is filtered,
\bea
V_0 \subset V_1 \subset \dots \subset V_k \subset \dots \subset V ,
\eea
the vector subspace $V_k \subset V$ consists of  knot invariants of order
not greater than $k$. We have $V_k \cdot V_l \subset V_{k+l}$. 

The subspace $V_0$ is of
dimension $1$ and consists of invariants taking the same value
on all knots. It is known that $V_1 = V_0$, \ $\dim V_2/V_1 = 
\dim V_3/V_2 = 1$. The generator in $V_2/V_1$ is given by the knot
invariant $x$ of order $2$ which takes value $0$ on the trivial knot and
value $8$ on the trefold. 
The generator in $V_3/V_2$ is given by the knot
invariant $y$ of order $3$ which takes value $0$ on the trivial knot, takes
value $24$ on the trefold, and takes value $-24$ on its  mirror image. 
Those conditions
determine $x$ and $y$ uniquely,  see for example \cite{L}.

It is known that the  space  $V_k$ has finite dimension 
fast growing with  $k$, see  for example \cite{CD, D, Z}. 




By definition the algebra $V$ is an
algebra of  certain special functions on the set $K$ of
all knots in $\R^3$ considered up to isotopy.
 
Let $T \subset K$ be the subset of toric knots of type $(n,2)$,
$n = \pm 1, \pm 3, \dots$.
Here $(1,2)$ and $(-1,2)$  
denote the trivial knot, 
$(3,2)$ is the trefoil, $(-3,2)$  its mirror image,  and so on.

Consider the algebra $A$ of functions on $T$, which is the restriction of
$V$ to $T$, i.e.  $A = V|_T$. The algebra $A$ is filtered,
\bea
A_0 \subset A_1 \subset \dots \subset A_k \subset \dots \subset A ,
\eea
where $A_k = V_k|_{T}$ for any $k$. Our goal is to describe $A$.

Let $X \in A_2$ and $Y \in A_3$ be the image of $x$ and $y$, 
respectively, under the natural projection $V \to A$.


\bigskip

\noindent
{\bf Theorem.}
{\it The algebra $A$ is generated by $X$ and $Y$ and is isomorphic to
the algebra $\Q[X,Y]/(X^3 + X^2 - Y^2)\Q[X,Y]$,\ where
$(X^3 + X^2 - Y^2)\Q[X,Y]\ \subset \ \Q[X,Y]$ 
is the ideal generated by the polynomial $X^3 + X^2 - Y^2$. 
We have $\dim A_0 = 1$, $\dim A_1/A_0 = 0$,
$\dim A_k/A_{k-1} = 1$ for $k > 1$. 
The generator in $ A_{2l}/A_{2l-1}$ is given by $X^l$
and the 
generator in $ A_{2l+1}/A_{2l}$ is given by $X^{l-1}Y$
for all $l > 0$. }

\bigskip

{\it Proof.} Denote by $\Zo$ the set of all odd integers.
For   $n\in \Zo$ denote by $[n]$ the  
torus knot of type $(n,2)$. 

An element $f \in A$ defines  a function
$ \Zo \to \Q$, $ n \mapsto f( [n] )$, and is uniquely
determined by that function. Thus $A$ can be considered
as an algebra of certain functions on $\Zo$.






\bigskip

\noindent
{\bf Lemma. }
{\it
\begin{enumerate}
\item[$\bullet$]
 If $f : \Zo \to \Q$ belongs to $A_k$ for some $k$, 
then $f$ is a polynomial of degree not greater than $k$.
\item[$\bullet$]
 If $f : \Zo \to \Q$ belongs to $A$, then $f(1) = f(-1)$.
\end{enumerate}
}
\hfill
$\square$

\bigskip

The lemma is a direct corollary of definitions.

\bigskip



We have $X : \Zo \to \Q,\ n \mapsto n^2-1$, and 
$Y : \Zo \to \Q, \ n \mapsto n^3-n$. This  gives the relation
$Y^2 = X^3 + X^2$. 

It is easy to see that 
all polynomials $ f : \Zo \to \Q$ with  property
$f(1) = f(-1)$ are linear combinations of monomials $X^l$ and $X^{l-1}Y$
of degree $2l$ and $2l+1$ respectively. 

The theorem is proved.
\hfill
$\square$

\bigskip


\noindent
{\bf Remarks.}
\begin{enumerate}
\item[$\bullet$]
After this paper had been written, S. Chmutov informed us about 
paper \cite{T}, where R. Trapp in particular shows that any element 
$f \in A$ is a polynomial function on $\Zo$ and $f$ can be written as a polynomial
in $X$ and $Y$.


\item[$\bullet$]  
S. Chmutov informed us that the shapes, similar to the shape of
our curve $Y^2 = X^3 + X^2$, appeared in \cite{W}, where
S. Willerton discusses statistics of points $(x(k),y(k)) \in \Q^2$
for arbitrary knots $k$.




\item[$\bullet$] According to our theorem the algebra $V|_T$ is isomorphic to the
algebra of regular functions on the affine curve $Y^2 = X^3 + X^2$.
One may wander what kind of algebraic varieties one obtains considering restrictions
of $V$ to other reasonable subsets of the set of all knots.





\end{enumerate}


\bigskip


\begin{thebibliography}{99}                                      

                                
\bibitem[B]{B} {\sc D. Bar-Natan,} 
{\it On the Vassiliev knot invariants}, Topology, {\bf 34} (1995), 
423--472.

\bibitem[CDL]{CDL} 
{\sc  S. Chmutov, S. Duzhin, and S. Lando} 
{\it Vassiliev Knot Invariants I. Introduction}, 
Adv. in Soviet Math. {\bf 21} Singularities and Curves, V.I.Arnold ed. 
(1994) 117--126.

\bibitem[CD]{CD} 
{\sc  S. Chmutov and S. Duzhin,} {\it 
An upper bound for the number of Vassiliev knot invariants}, 
Journal of Knot Theory and its Ramifications, {\bf 3} (1994), 141--151.

\bibitem[D]{D} {\sc O. Dasbach,}
{\it On the Combinatorial Structure of Primitive Vassiliev Invariants III
-- A Lower Bound}, Communications in Contemporary Mathematics,
{\bf 2} (2000), No. 4, 579-590.

\bibitem[L]{L} 
{\sc J. Lannes,} {\it Sur les invariants de Vassiliev de degre\'e inferieur
ou \'egal \`a 3,} L'En\-seignement Math\'ematique, {\bf 39} (1993), 295--316.

\bibitem[T]{T} {\sc R. Trapp,} {\it Twist Sequences and Vassiliev Invariants},
Journal of Knot Theory and its Ramifications, {\bf 3} (1994), No. 3, 391--405.




\bibitem[Ty]{Ty} {\sc S. Tyurina,} {\it Diagram formulas of Viro-Polyak type
and the Kontsevich integral for $(2,n)$-torus knots,} Proc. Rokhlin Memorial 
Conference, Euler Institute, Saint-Petersburg, August 1999.  

\bibitem[Z]{Z} {\sc D. Zagier,} {\it Vassiliev Invariants and a Strange Identity
Related to the Dedekind Eta-Function}, Topology {\bf 40} (5)
(2001), 945--960.


\bibitem[W]{W} {\sc S. Willerton,} {\it On the First Two 
Vassiliev Invariants,} math.GT/0104061, 1--11.



\end{thebibliography}






\end{document}





Consider the $\Q$-vector space  $K$ with basis labeled by
isotopy classes of knots. 

Let $n$ be a natural number.
Denote by $K^*_n$ the $\Q$-vector space of all linear functions on $K$
of order not greater than $n$. Recall that a linear function $f$
has order not greater than $n$ if it takes the zero value on any linear
combination of $2^{n+1}$-knots constructed as follows.

.....

.....

$$
Picture \ 1\ .
$$

.....

.....


The space $K^*_n$ has finite dimension 
exponentially growing with  $n$, see  \cite{CD}. 

Denote by $T$ the vector subspace in $K$ generated by torus knots of
type $(2l+1, 2)$, $l\in \Z$. Here $(1,2)$ and $(-1,2)$  
denote the trivial knot, 
$(3,2)$ is the trefoil, $(-3,2)$ is its mirrow image,  and so on, see Picture 2.
$$
Picture \ 2 \ .
$$
Denote by $T^*_n$ the restriction to $T$ of linear functions of order not
greater than $n$, \linebreak
$T^*_n = \{ f|_{T}\ |\ f \in K^*_n\}$.




\noindent
{\bf Theorem 1.}
{\it Let $n > 1$. Then the
vector space $T^*_n$ has dimension $n$.
The linear functions $t_i : T \to \Q$, $i = 0, 2, 3, \dots , n$, defined below
in Section 2 form a basis in $T^*_n$.}



\bigskip

\noindent
{\bf 2.  Definition of basis invariants. }
For $l\in \Z$ denote by $[2l+1]$ the  vector in $T$ corresponding to
the torus knot of type $(2l+1,2)$. The basis in $T$ is formed by the vectors
$[2l+1]$, $l = 0, 1, \pm 2, \pm 3, ...$.

For $i = 0, 2, 3, \dots , k$, define $t_i\in T^*_n$.

Set $t_0( [2l+1] ) = 1$ for all $l$.

If $i\in \{2, \dots , k\}$ and $t$ is odd, set
$t_i ( [2l+1] ) = ???$ for all $l$.

If $i\in \{2, \dots , k\}$ and $t$ is even, set
$t_i ( [2l+1] ) = ???$ for all $l$.

\bigskip

\noindent
{\bf 3.  Proof of Theorem 1. }


\bigskip

\noindent
{\bf Corollary.}
{\it If  $t \in T^*_n$, then $t$ is uniquely defined by its values on
the torus knots $[2l+1]$ with $|l| \leq n$. }




Consider the $\Q$-vector space  $K$ with basis labeled by
isotopy classes of knots. 

Let $n$ be a natural number.
Denote by $K^*_n$ the $\Q$-vector space of all linear functions on $K$
of order not greater than $n$. Recall that a linear function $f$
has order not greater than $n$ if it takes the zero value on any linear
combination of $2^{n+1}$-knots constructed as follows.

.....

.....

$$
Picture \ 1\ .
$$

.....

.....


The space $K^*_n$ has finite dimension 
exponentially growing with  $n$, see  \cite{CD}. 

Denote by $T$ the vector subspace in $K$ generated by torus knots of
type $(2l+1, 2)$, $l\in \Z$. Here $(1,2)$ and $(-1,2)$  
denote the trivial knot, 
$(3,2)$ is the trefoil, $(-3,2)$ is its mirror image,  and so on, see Picture 2.
$$
Picture \ 2 \ .
$$
Denote by $T^*_n$ the restriction to $T$ of linear functions of order not
greater than $n$, \linebreak
$T^*_n = \{ f|_{T}\ |\ f \in K^*_n\}$.




\noindent
{\bf Theorem 1.}
{\it Let $n > 1$. Then the
vector space $T^*_n$ has dimension $n$.
The linear functions $t_i : T \to \Q$, $i = 0, 2, 3, \dots , n$, defined below
in Section 2 form a basis in $T^*_n$.}



\bigskip

\noindent
{\bf 2.  Definition of basis invariants. }
For $l\in \Z$ denote by $[2l+1]$ the  vector in $T$ corresponding to
the torus knot of type $(2l+1,2)$. The basis in $T$ is formed by the vectors
$[2l+1]$, $l = 0, 1, \pm 2, \pm 3, ...$.

For $i = 0, 2, 3, \dots , k$, define $t_i\in T^*_n$.

Set $t_0( [2l+1] ) = 1$ for all $l$.

If $i\in \{2, \dots , k\}$ and $t$ is odd, set
$t_i ( [2l+1] ) = ???$ for all $l$.

If $i\in \{2, \dots , k\}$ and $t$ is even, set
$t_i ( [2l+1] ) = ???$ for all $l$.

\bigskip

\noindent
{\bf 3.  Proof of Theorem 1. }


\bigskip

\noindent
{\bf Corollary.}
{\it If  $t \in T^*_n$, then $t$ is uniquely defined by its values on
the torus knots $[2l+1]$ with $|l| \leq n$. }




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\bibitem[CD]{CD} 
{\sc  S. Chmutov and S. Duzhin,} {\it Vassiliev Knot Invariants I. Introduction}, 
preprint.
                                                  
\bibitem[B]{Bar95} 
 {\sc D. Bar-Natan,} {\it On the Vassiliev knot invariants}, -- Topology. {\bf 34}, 1995, 
423--472.

\bibitem[BGRT]{Bar97}  {\sc D. Bar-Natan, S.Garoufalidis, L.Rozansky,
and D. Thurston,} {\it Wheels, wheeling, and the Kontsevich
integral of the unknot}, arXiv:q-alg/9703025.

\bibitem[CV]{CV}
{\sc S. Chmutov and A. Varchenko,}
{\it Remarks on the Vassiliev Knot Invariants Coming from $sl_2.$}
-- Topology. {\bf 36}, No.1, 1996, 153--178.

\bibitem[K]{K}
{\sc M. Kontsevich,} {\it Vassiliev's knot invariants, }
 Adv. in Sov. Math. {\bf 16}, part 2, 1993, 137--150.


\bibitem[LM]{LM}
{\sc T.Q.T. LE and J. MURAKAMI,} {\it Parallel version of the universal 
Vassiliev-Kontsevich invariant.} -- J.
Pure and Appl. Alg. {\bf 121}, 1997, 271--291.

\bibitem[T]{T} {\sc D. Thurston,} {\it Wheeling: a diagrammatic analogue
of the Duflo isomorphism}, Ph.D.thesis, arXiv:math.QA/0006083.

















math0402185.extracted_bib
\begin{thebibliography}{99}                                      

                                
\bibitem[B]{B} {\sc D. Bar-Natan,} 
{\it On the Vassiliev knot invariants}, Topology, {\bf 34} (1995), 
423--472.

\bibitem[CDL]{CDL} 
{\sc  S. Chmutov, S. Duzhin, and S. Lando} 
{\it Vassiliev Knot Invariants I. Introduction}, 
Adv. in Soviet Math. {\bf 21} Singularities and Curves, V.I.Arnold ed. 
(1994) 117--126.

\bibitem[CD]{CD} 
{\sc  S. Chmutov and S. Duzhin,} {\it 
An upper bound for the number of Vassiliev knot invariants}, 
Journal of Knot Theory and its Ramifications, {\bf 3} (1994), 141--151.

\bibitem[D]{D} {\sc O. Dasbach,}
{\it On the Combinatorial Structure of Primitive Vassiliev Invariants III
-- A Lower Bound}, Communications in Contemporary Mathematics,
{\bf 2} (2000), No. 4, 579-590.

\bibitem[L]{L} 
{\sc J. Lannes,} {\it Sur les invariants de Vassiliev de degre\'e inferieur
ou \'egal \`a 3,} L'En\-seignement Math\'ematique, {\bf 39} (1993), 295--316.

\bibitem[T]{T} {\sc R. Trapp,} {\it Twist Sequences and Vassiliev Invariants},
Journal of Knot Theory and its Ramifications, {\bf 3} (1994), No. 3, 391--405.




\bibitem[Ty]{Ty} {\sc S. Tyurina,} {\it Diagram formulas of Viro-Polyak type
and the Kontsevich integral for $(2,n)$-torus knots,} Proc. Rokhlin Memorial 
Conference, Euler Institute, Saint-Petersburg, August 1999.  

\bibitem[Z]{Z} {\sc D. Zagier,} {\it Vassiliev Invariants and a Strange Identity
Related to the Dedekind Eta-Function}, Topology {\bf 40} (5)
(2001), 945--960.


\bibitem[W]{W} {\sc S. Willerton,} {\it On the First Two 
Vassiliev Invariants,} math.GT/0104061, 1--11.



\end{thebibliography}