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

Garoufalidis, Stavros¹ ; Lan, Yueheng²

¹School of Mathematics, Georgia Institute of Technology, Atlanta GA 30332-0160, USA
²School of Physics, Georgia Institute of Technology, Atlanta GA 30332-0160, USA

Algebraic and Geometric Topology, Tome 5 (2005) no. 1, pp. 379-403

Résumé

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.

Détail

DOI : 10.2140/agt.2005.5.379

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

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