Volume 5, issue 1 (2005)

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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
On the Mahler measure of Jones polynomials under twisting

Abhijit Champanerkar and Ilya Kofman

Algebraic & Geometric Topology 5 (2005) 1–22

arXiv: math.GT/0404236

Abstract

We show that the Mahler measures of the Jones polynomial and of the colored Jones polynomials converge under twisting for any link. Moreover, almost all of the roots of these polynomials approach the unit circle under twisting. In terms of Mahler measure convergence, the Jones polynomial behaves like hyperbolic volume under Dehn surgery. For pretzel links P(a1,,an), we show that the Mahler measure of the Jones polynomial converges if all ai , and approaches infinity for ai = constant if n , just as hyperbolic volume. We also show that after sufficiently many twists, the coefficient vector of the Jones polynomial and of any colored Jones polynomial decomposes into fixed blocks according to the number of strands twisted.

Keywords
Jones polynomial, Mahler measure, Temperley–Lieb algebra, hyperbolic volume
Mathematical Subject Classification 2000
Primary: 57M25
Secondary: 26C10
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Publication
Received: 13 October 2004
Revised: 6 November 2004
Accepted: 7 December 2004
Published: 5 January 2005
Authors
Abhijit Champanerkar
Department of Mathematics
Barnard College
Columbia University
3009 Broadway
New York NY 10027
USA
Ilya Kofman
Department of Mathematics
Columbia University
2990 Broadway
New York NY 10027
USA