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

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.