A volumish theorem for the Jones polynomial of alternating knots

Oliver T. Dasbach and Xiao-Song Lin

Vol. 231 (2007), No. 2, 279–291

DOI: 10.2140/pjm.2007.231.279

Abstract

The Volume Conjecture claims that the hyperbolic volume of a knot is determined by the colored Jones polynomial.

Here we prove a “Volumish Theorem” for alternating knots in terms of the Jones polynomial, rather than the colored Jones polynomial: The ratio of the volume and certain sums of coefficients of the Jones polynomial is bounded from above and from below by constants.

Furthermore, we give experimental data on the relation of the growths of the hyperbolic volume and the coefficients of the Jones polynomial, both for alternating and nonalternating knots.

Keywords

Jones polynomial, hyperbolic volume, alternating knots, volume conjecture

Mathematical Subject Classification 2000

Primary: 57M25

Milestones

Received: 18 January 2006

Accepted: 31 October 2006

Published: 1 June 2007