The Volume Conjecture for small angles.
by
Stavros Garoufalidis
School of Mathematics, Georgia Institute of Technology
Coauthors: Thang TQ Le

The Generalized Hyperbolic Volume Conjecture (GHVC) states that the n-th colored Jones polynomial, evaluated at exp(2 pi i a/n), is a sequence of complex numbers that grows exponentially. Moroever, the exponential growth rate is proportional to the volume of the corresponding Dehn filling. We prove two statements: (a) the limsup in the GHVC is finite for all knots and all a. (b) for every knot K there exists a positive angle a(K) such that the GHVC holds for a in [0, a(K)). The proofs of these statements use elementary properties of state sum formulas for the colored Jones polynomial, and its recursion and its cyclotomic expansion. Finally, we mention that our upper bounds for the limsup and the volume (linear in terms of the number of crossings of a knot) are asymtptoically optimal. The latter uses work of Agol-Storm-W.Thurston, which is based on work of Perelman.

Date received: February 8, 2005