A Remark on the sl2 Approximation of the Kontsevich Integral of the Unknot

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 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 \(I'(K) = 1 + \sum {_{n = 1}^\infty } h^n E_n\) 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)).