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Zapiski Nauchnykh Seminarov POMI, 2003, Volume 299, Pages 30–37
(Mi znsl1030)
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A remark on the $sl_2$ approximation of the Kontsevich integral of the unknot
A. N. Varchenkoa, S. Tyurinab a Department of Mathematics, University of North Carolina at Chapel Hill
b Max Planck Institute for Mathematics
Abstract:
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.
Received: 19.11.2001
Citation:
A. N. Varchenko, S. Tyurina, “A remark on the $sl_2$ approximation of the Kontsevich integral of the unknot”, Geometry and topology. Part 8, Zap. Nauchn. Sem. POMI, 299, POMI, St. Petersburg, 2003, 30–37; J. Math. Sci. (N. Y.), 131:1 (2005), 5270–5274
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https://www.mathnet.ru/eng/znsl1030 https://www.mathnet.ru/eng/znsl/v299/p30
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