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

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

1996-04-03

arxiv.org/abs/q-alg/9604005v2

Mathematics

Quantum Algebra

31 pages, LaTeX (some misprints corrected, references added)

Scientific paper

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.

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