Algebraic and Geometric Topology 5 (2005), paper no. 1, pages 1-22.

On the Mahler measure of Jones polynomials under twisting

Abhijit Champanerkar, Ilya Kofman

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(a_1,...,a_n), we show that the Mahler measure of the Jones polynomial converges if all a_i tend to infinity, and approaches infinity for a_i = constant if n tend to infinity, 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

AMS subject classification. Primary: 57M25. Secondary: 26C10.

DOI: 10.2140/agt.2005.5.1

E-print: arXiv:math.GT/0404236

Submitted: 13 October 2004. (Revised: 6 November 2004.) Accepted: 7 December 2004. Published: 5 January 2005.

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Abhijit Champanerkar, Ilya Kofman
Department of Mathematics, Barnard College, Columbia University
3009 Broadway, New York, NY 10027, USA
and
Department of Mathematics, Columbia University
2990 Broadway, New York, NY 10027, USA

Email: abhijit@math.columbia.edu, ikofman@math.columbia.edu

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