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

TL;DR

This paper conjectures a comprehensive all-order asymptotic expansion for the colored Jones polynomial of hyperbolic knot complements, incorporating non-perturbative corrections via topological recursion and character varieties.

Contribution

It introduces a non-perturbative framework for the asymptotics of knot invariants using topological recursion and character varieties, extending previous perturbative approaches.

Findings

Confirmed the conjecture for the figure-eight knot.

Validated the approach on the once-punctured torus bundle.

Provided a heuristic argument for torus knots using matrix models.

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 = exp(2u/N)) when N goes to infinity. Our conjecture claims that the asymptotic expansion of the colored Jones polynomial is a the formal wave function of an integrable system whose semiclassical spectral curve S would be the SL_2(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 S, but the expansion is still in powers of 1/N due to the special properties of A-polynomials. We provide a detailed check for the figure-eight knot and the once-punctured torus bundle L^2R. We also present a heuristic argument inspired from the case of torus knots, for which knot invariants can be computed from a matrix model.