dorsal/arxiv
View SchemaThe Universal R-Matrix, Burau Representaion and the Melvin-Morton Expansion of the Colored Jones Polynomial
| Authors | L. Rozansky |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9604005 |
| URL | https://arxiv.org/abs/q-alg/9604005 |
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.
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"abstract": "P. Melvin and H. Morton studied the expansion of the colored Jones polynomial\nof a knot in powers of q-1 and color. They conjectured an upper bound on the\npower of color versus the power of q-1. They also conjectured that the bounding\nline in their expansion generated the inverse Alexander-Conway polynomial.\nThese conjectures were proved by D. Bar-Natan and S. Garoufalidis.\n We have conjectured that other `lines\u0027 in the Melvin-Morton expansion are\ngenerated by rational functions with integer coefficients whose denominators\nare powers of the Alexander-Conway polynomial. Here we prove this conjecture by\nusing the R-matrix formula for the colored Jones polynomial and presenting the\nuniversal R-matrix as a `perturbed\u0027 Burau matrix.",
"arxiv_id": "q-alg/9604005",
"authors": [
"L. Rozansky"
],
"categories": [
"q-alg",
"math.QA"
],
"title": "The Universal R-Matrix, Burau Representaion and the Melvin-Morton Expansion of the Colored Jones Polynomial",
"url": "https://arxiv.org/abs/q-alg/9604005"
},
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