Algebraic & Geometric Topology

On the Mahler measure of Jones polynomials under twisting

Abhijit Champanerkar and Ilya Kofman

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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.

Article information

Algebr. Geom. Topol., Volume 5, Number 1 (2005), 1-22.

Received: 13 October 2004
Revised: 6 November 2004
Accepted: 7 December 2004
First available in Project Euclid: 20 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 26C10: Polynomials: location of zeros [See also 12D10, 30C15, 65H05]

Jones polynomial Mahler measure Temperley–Lieb algebra hyperbolic volume


Champanerkar, Abhijit; Kofman, Ilya. On the Mahler measure of Jones polynomials under twisting. Algebr. Geom. Topol. 5 (2005), no. 1, 1--22. doi:10.2140/agt.2005.5.1.

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