Algebraic and Geometric Topology 5 (2005), paper no. 17, pages 379-403.

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

Stavros Garoufalidis, Yueheng Lan

Abstract. The Volume Conjecture loosely 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

AMS subject classification. Primary: 57N10. Secondary: 57M25.

DOI: 10.2140/agt.2005.5.379

E-print: arXiv:math.GT/0412331

Submitted: 16 December 2004. (Revised: 21 April 2005.) Accepted: 6 May 2005. Published: 22 May 2005.

Notes on file formats

Stavros Garoufalidis, Yueheng Lan
School of Mathematics, Georgia Institute of Technology
Atlanta, GA 30332-0160, USA
and
School of Physics, Georgia Institute of Technology
Atlanta, GA 30332-0160, USA
Email: stavros@math.gatech.edu, gte158y@mail.gatech.edu
URL: http://www.math.gatech.edu/~stavros, http://cns.physics.gatech.edu/~y-lan

AGT home page

EMIS/ELibM Electronic Journals

Outdated Archival Version

These pages are not updated anymore. They reflect the state of 21 Apr 2006. For the current production of this journal, please refer to http://msp.warwick.ac.uk/.