The UniversalR-Matrix, Burau Representation, and the Melvin-Morton Expansion of the Colored Jones Polynomial

Abstract

P. Melvin and H. Morton [9] studied the expansion of the colored Jones polynomial of a knot in powers ofq-1 and color. They conjectured an upper bound on the power of color versus the power of -1. They also conjectured that the bounding line in their expansion generated the inverse Alexander-Conway polynomial. These conjectures were proved by D. Bar-Natan and S. Garoufalidis [1]. We have conjectured [12] that other 'lines' in the Melvin-Morton expansion are generated by rational functions with integer coefficients whose denominators are powers of the Alexander-Conway polynomial. Here we prove this conjecture by using theR-matrix formula for the colored Jones polynomial and presenting the universalR-matrix as a 'perturbed' Burau matrix. © 1998 Academic Press.