For a fixed Turaev shadow Jones-Vassiliev invariants depend polynomially on the gleams

Abstract

We use Turaev's technique of shadows and gleams to parametrize the set of all knots in S 3 with the same Hopf projection. We show that the Vassiliev invariants arising from the Jones polynomial Jt (K) are polynomials in the gleams, i.e., for $n\ge 2$ , the n-th order Vassiliev invariant un, defined by $J_{e^x}(K)={\sum }_{n=0}^{\infty }{u}_n(K)x^n$ , is a polynomial of degree 2n in the gleams.