On links with cyclotomic Jones polynomials

Champanerkar, Abhijit; Kofman, Ilya

doi:10.2140/agt.2006.6.1655

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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 ${L_{n}}$ is any infinite sequence of links with twist number $τ (L_{n})$ and with cyclotomic Jones polynomials of increasing span, then $lim sup τ (L_{n}) = \infty$ . 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 2000

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

Volume 6, issue 4 (2006)