On the volume conjecture for small angles

Given a knot in 3-space, one can associate a sequence of Laurrent polynomials, whose $n$th term is the $n$th colored Jones polynomial. The Generalized Volume Conjecture states that the value of the $n$-th colored Jones polynomial at $\exp(2 \pi i \a/n)$ is a sequence of complex numbers that grows exponentially, for a fixed real angle $\a$. Moreover the exponential growth rate of this sequence is proportional to the volume of the 3-manifold obtained by $(1/\a,0)$ Dehn filling. In this paper we will prove that (a) for every knot, the limsup in the hyperbolic volume conjecture is finite and bounded above by an exponential function that depends on the number of crossings. (b) Moreover, for every knot $K$ there exists a positive real number $\a(K)$ (which depends on the number of crossings of the knot) such that the Generalized Volume Conjecture holds for $\a \in [0, \a(K))$. Finally, we point out that a theorem of Agol-Storm-W.Thurston proves that the bounds in (a) are optimal, given by knots obtained by closing large chunks of the weave.