Asymptotic of colored Jones polynomials, topological recursion and theta functions

I will start with a short review of the generalized volume conjecture (cf. Pr. Murakami Oberseminar last week) about asymptotics of the colored Jones polynomial of a knot. Then, I will present (with illustration for the figure eight-knot) a conjecture aiming at computing the all-order asymptotics of the colored Jones polynomial of hyperbolic knots, solely in terms of algebraic geometry on the character variety. If time is available, I shall discuss the arithmetic properties of a specialization of our construction, which is conjectured to match the expansion of the Kashaev invariant. The talk is based on a joint work with Bertrand Eynard.