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On
the Mahler measure of Jones polynomials under twisting
Abhijit Champanerkar and Ilya Kofman |
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Algebraic & Geometric Topology 5 (2005) 1–22
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DOI: 10.2140/agt.2005.5.1
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arXiv: math.GT/0404236 |
Abstract |
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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 , we show that the Mahler measure of the Jones polynomial converges if all , and approaches infinity for constant if , 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
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Mathematical Subject Classification 2000
Primary: 57M25
Secondary: 26C10
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References |
Forward citations |
Publication
Received: 13 October 2004
Revised: 6 November 2004
Accepted: 7 December 2004
Published: 5 January 2005
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Authors |
| Abhijit Champanerkar | |
| Department of Mathematics Barnard College Columbia University 3009 Broadway New York NY 10027 USA |
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| Ilya Kofman | |
| Department of Mathematics Columbia University 2990 Broadway New York NY 10027 USA |
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