Weights of Feynman diagrams, link polynomials and Vassiliev knot invariants

Abstract

We prove that the construction of Vassiliev invariants by expanding the link polynomials P g,V (q, q −1 ) at the point q=1 is equivalent to the construction of Vassiliev invariants from Chern-Simons perturbation theory. In both constructions a simple Lie algebra g and an irreducible representation V of g should be specified. We give an example of a Vassiliev invariant of order six which cannot be obtained by these constructions if we restrict ourselves to simple Lie algebras and do not allow semisimple ones. The explicit description of primitive elements in the Kontsevich Hopf algebra is given.