#### 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.

Notes on file formats
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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