Experimental evidence for the Volume Conjecture for the simplest hyperbolic non-2–bridge knot

Garoufalidis, Stavros; Lan, Yueheng

doi:10.2140/agt.2005.5.379

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Experimental evidence for the Volume Conjecture for the simplest hyperbolic non-2–bridge knot

Stavros Garoufalidis and Yueheng Lan

Algebraic & Geometric Topology 5 (2005) 379–403

DOI: 10.2140/agt.2005.5.379

arXiv: math.GT/0412331

Abstract

Loosely speaking, the Volume Conjecture states that the limit of the $n$ th colored Jones polynomial of a hyperbolic knot, evaluated at the primitive complex $n$ th root of unity is a sequence of complex numbers that grows exponentially. Moreover, the exponential growth rate is proportional to the hyperbolic volume of the knot. We provide an efficient formula for the colored Jones function of the simplest hyperbolic non-2–bridge knot, and using this formula, we provide numerical evidence for the Hyperbolic Volume Conjecture for the simplest hyperbolic non-2–bridge knot.

Keywords

knots, $q$–difference equations, asymptotics, Jones polynomial, Hyperbolic Volume Conjecture, character varieties, recursion relations, Kauffman bracket, skein module, fusion, SnapPea, m082

Mathematical Subject Classification 2000

Primary: 57N10

Secondary: 57M25

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Publication

Received: 16 December 2004

Revised: 21 April 2005

Accepted: 6 May 2005

Published: 22 May 2005

Authors

Stavros Garoufalidis
School of Mathematics Georgia Institute of Technology Atlanta GA 30332-0160 USA
http://www.math.gatech.edu/~stavros/
Yueheng Lan
School of Physics Georgia Institute of Technology Atlanta GA 30332-0160 USA
http://cns.physics.gatech.edu/~y-lan/

Volume 5, issue 1 (2005)