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.