The Universal R-Matrix, Burau Representaion and the Melvin-Morton Expansion of the Colored Jones Polynomial

TL;DR

This paper proves a conjecture relating the Melvin-Morton expansion of the colored Jones polynomial to rational functions generated by the universal R-matrix, connecting knot invariants with algebraic structures.

Contribution

It demonstrates that lines in the Melvin-Morton expansion are generated by rational functions with denominators as powers of the Alexander-Conway polynomial, using the R-matrix and Burau matrix.

Findings

Proved the conjecture about the structure of the Melvin-Morton expansion.

Connected the universal R-matrix to the Burau representation.

Established a new algebraic interpretation of the colored Jones polynomial.

Abstract

P. Melvin and H. Morton studied the expansion of the colored Jones polynomial of a knot in powers of q-1 and color. They conjectured an upper bound on the power of color versus the power of q-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. We have conjectured 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 the R-matrix formula for the colored Jones polynomial and presenting the universal R-matrix as a `perturbed' Burau matrix.