A REMARK ON THE sl2 APPROXIMATION OF THE KONTSEVICH INTEGRAL OF THE UNKNOT


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Abstract

The Kontsevich integral of a knot K is a sum I(K) = 1 + ∑ ∞n = 1hn ∑ aD D over all chord diagrams with suitable coefficients. Here An is the space of chord diagrams with n chords. A simple explicit formula for the coefficients aD is not known even for the unknot. Let E1, E2,… be elements of A = ⊕n An. Say that the sum I1 (K)=1 + ∑ ∞ n=1 hnEn is an sl2 approximation of the Kontsevich integral if the values of the sl2 weight system Wsl2 on both sums are equal: Wsl2 (I(K)) = Wsl2 (I'(K)).For any natural n fix points a1,…, a2n on a circle. For any permutation α ε S2n of 2n elements, one defines the chord diagram D(α) with n chords as the diagram with chords formed by pairs aα (2i-1) and aα(2i), i = 1, …,n. It is shown that $$ is an sl2 approximation of the Kontsevich integral of the unknot. Bibliography: 6 titles.

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