On the Melvin-Morton-Rozansky conjecture
A conjecture of Melvin and Morton is proved which implies that the Alexander-Conway polynomial of a knot can be read from some of the coefficients of the Jones polynomials of cables of the knot (i.e., coefficients of the ``colored'' Jones polynomial). Rozansky gave a ``non-mathematical proof'' of the conjecture using the path integral interpretation of the Jones polynomial given by Witten; his argument is outlined in the introduction of the present paper. The main tools of the present paper are Vassiliev invariants and weight systems associated to chord diagrams. It is well known that the coefficients of both the Conway and the Jones polynomials are Vassiliev invariants. In general, Vassiliev invariants are not determined by their weight systems. However, if the invariant comes from a Lie algebra, then it is determined by its weight system. As this is the case for all invariants appearing in the conjecture, the conjecture is proved combinatorially on the level of weight systems. Also, a generalization of the Melvin-Morton-Rozansky conjecture is proved for knot invariants coming from arbitrary semisimple Lie algebras.
scientific article; zbMATH DE number 885522