On the Melvin-morton-rozansky Conjecture

DROR BAR-NATAN AND STAVROS GAROUFALIDIS This is a pre-preprint. Corrections, suggestions, reservations and donations are more than welcome! Abstract. We prove a conjecture stated by Melvin and Morton (and elucidated further by Rozansky) saying that the Alexander-Conway polynomial of a knot can be read from some of the coeecients of the Jones polynomials of cables of that knot (i.e., coeecients of the \colored" Jones polynomial). We rst reduce the problem to the level of weight systems using a general principle, which may be of some independent interest, and which sometimes allows to deduce equality of Vassiliev invariants from the equality of their weight systems. We then prove the conjecture combinatorially on the level of weight systems. Finally, we prove a generalization of the Melvin-Morton-Rozansky (MMR) conjecture to knot invariants coming from arbitrary semi-simple Lie algebras. As side beneets we discuss a relation between the Conway polynomial and immanants and a curious formula for the weight system of the colored Jones polynomial.