A~remark on the~$sl_2$ approximation of the Kontsevich integral of the unknot

A. N. Varchenko ; S. Tyurina

Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 8, Tome 299 (2003), pp. 30-37

Résumé

The Kontsevich integral of a knot $K$ is a sum $I(K)=1+\sum_{n=1}^\infty h^n\sum_{D\in A_n}a_D D$ over all chord diagrams with suitable coefficients. Here $A_n$ is the space of chord diagrams with $n$ chords. A simple explicit formula for the coefficients $a_D$ is not known even for the unknot. Let $E_1,E_2,\dots$ be elements of $A=\bigoplus_{n}A_n$. Say that the sum $I'(K)=1+\sum_{n=1}^\infty h^n E_n$ is an $sl_2$ approximation of the Kontsevich integral if the values of the $sl_2$ weight system $W_{sl_2}$ on both sums are equal: $W_{sl_2}(I(K))=W_{sl_2}(I'(K))$. For any natural n fix points $a_1,\dots,a_2n$ on a circle. For any permutation $\sigma\in S_{2n}$ of $2n$ elements, one defines the chord diagram $D(\sigma)$ with $n$ chords as the diagram with chords formed by pairs $(a_{\sigma(2i-1)} and a_{\sigma(2i)})$, $i=1,\dots,n$. It is shown that $$ 1+\sum_{n=1}^\infty\frac{h^{2n}}{2^n(2n)!(2n+1)!}\sum_{\sigma\in S_{2n}}D(\sigma) $$ is an $sl_2$ approximation of the Kontsevich integral of the unknot.