Volume 5, issue 1 (2005)

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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
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

arXiv: math.GT/0412331

Abstract

Loosely speaking, the Volume Conjecture states that the limit of the nth colored Jones polynomial of a hyperbolic knot, evaluated at the primitive complex nth 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
References
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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/