The volume conjecture for small angles. (Stavros Garoufalidis)


Given a knot in 3-space, one can associate a sequence J(n) of Laurent polynomials in a variable q, where n=1,2,3... The volume conjecture states that the evaluation of J(n) at exp(2 pi i a/n) is a sequence of complex numbers that grows exponentially. Moreover, the growth rate is proportional to the volume of a corresponding SL(2,C) representation of the knot complement. In joint work with Thang Le, we will give a proof of the volume conjecture for small a-angles. Moreover, we will prov the existence of asymptotic expansions to all orders in n, for small angles.