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On links with cyclotomic Jones polynomials

Abhijit Champanerkar and Ilya Kofman

Algebraic & Geometric Topology 6 (2006) 1655–1668

DOI: 10.2140/agt.2006.6.1655

arXiv: math.GT/0605631
Abstract

We show that if {Ln} is any infinite sequence of links with twist number τ(Ln) and with cyclotomic Jones polynomials of increasing span, then lim sup τ(Ln)=∞. This implies that any infinite sequence of prime alternating links with cyclotomic Jones polynomials must have unbounded hyperbolic volume. The main tool is the multivariable twist–bracket polynomial, which generalizes the Kauffman bracket to link diagrams with open twist sites.

Keywords

Jones polynomial, Mahler measure, twist sites, hyperbolic volume
Mathematical Subject Classification

Primary: 57M25

Secondary: 26C10
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Publication

Received: 5 June 2006
Accepted: 28 August 2006
Published: 14 October 2006
Authors
Abhijit Champanerkar
Department of Mathematics and Statistics
University of South Alabama
Mobile, AL 36688
USA
Ilya Kofman
Department of Mathematics
College of Staten Island
City University of New York
2800 Victory Boulevard
Staten Island, NY 10314
USA