1996/04/03 by Lev Rozansky, L. Rozansky, Rozansky, L. · 2 citations
Computer Science · Mathematics · #57M25 (Primary) 17B37 (Secondary) #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric and Algebraic Topology #Quantum Algebra (math.QA) #math.QA #msc:17B37 #msc:57M25 #q-alg #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.q-alg/9604005
31 pages, LaTeX (some misprints corrected, references added)
openalex publication_date 1996/04/03 · arxiv created 1996/05/17 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.