The Universal R-Matrix, Burau Representaion and the Melvin-Morton 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 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.
- Publication:
-
eprint arXiv:q-alg/9604005
- Pub Date:
- April 1996
- DOI:
- 10.48550/arXiv.q-alg/9604005
- arXiv:
- arXiv:q-alg/9604005
- Bibcode:
- 1996q.alg.....4005R
- Keywords:
-
- Mathematics - Quantum Algebra;
- 57M25 (Primary) 17B37 (Secondary)
- E-Print:
- 31 pages, LaTeX (some misprints corrected, references added)