The Universal RMatrix, Burau Representaion and the MelvinMorton Expansion 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 q1 and color. They conjectured an upper bound on the power of color versus the power of q1. They also conjectured that the bounding line in their expansion generated the inverse AlexanderConway polynomial. These conjectures were proved by D. BarNatan and S. Garoufalidis. We have conjectured that other `lines' in the MelvinMorton expansion are generated by rational functions with integer coefficients whose denominators are powers of the AlexanderConway polynomial. Here we prove this conjecture by using the Rmatrix formula for the colored Jones polynomial and presenting the universal Rmatrix as a `perturbed' Burau matrix.
 Publication:

eprint arXiv:qalg/9604005
 Pub Date:
 April 1996
 DOI:
 10.48550/arXiv.qalg/9604005
 arXiv:
 arXiv:qalg/9604005
 Bibcode:
 1996q.alg.....4005R
 Keywords:

 Mathematics  Quantum Algebra;
 57M25 (Primary) 17B37 (Secondary)
 EPrint:
 31 pages, LaTeX (some misprints corrected, references added)