All order asymptotics of hyperbolic knot invariants from non-perturbative topological recursion of A-polynomials

Abstract

We propose a conjecture to compute the all-order asymptotic expansion of the colored Jones polynomial of the complement of a hyperbolic knot, $J_N(q=e^{\frac{2u}{N}})$ when $N\to \infty$. Our conjecture claims that the asymptotic expansion of the colored Jones polynomial is a formal wave function of an integrable system whose semiclassical spectral curve $\mathcal{C}$ would be the ${\mathrm{SL}}_2(\mathbb{C})$ character variety of the knot (the A-polynomial), and is formulated in the framework of the topological recursion. It takes as starting point the proposal made recently by Dijkgraaf, Fuji and Manabe (who kept only the perturbative part of the wave function, and found some discrepancies), but it also contains the non-perturbative parts, and solves the discrepancy problem. These non-perturbative corrections are derivatives of Theta functions associated to $\mathcal{C}$. For a large class of knots, this expansion is still in powers of $1/N$ due to the special properties of A-polynomials. We provide a detailed check of our proposal for the figure-eight knot and the once-punctured torus bundle $L^{2}R$. We also present a heuristic argument inspired from the case of torus knots, for which knot invariants can be computed by a matrix model.