#### 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

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