The resurgent structure of quantum knot invariants

TL;DR

This paper explores the resurgent structure of quantum knot invariants derived from complex Chern-Simons theory, proposing conjectures about their asymptotic expansions, Stokes automorphisms, and connections to BPS state counting, supported by explicit examples.

Contribution

It introduces conjectures linking the resurgent properties of quantum knot invariants to $q$-series matrices and BPS state counting, with explicit theoretical and numerical evidence for specific knots.

Findings

Conjecture that quantum knot invariants' asymptotic series are resurgent functions with explicit Stokes automorphisms.

Identification of a connection between a matrix entry of these automorphisms and the 3D-index for hyperbolic knots.

Numerical verification of the conjectures for the $4_1$ and $5_2$ knots.

Abstract

The asymptotic expansion of quantum knot invariants in complex Chern-Simons theory gives rise to factorially divergent formal power series. We conjecture that these series are resurgent functions whose Stokes automorphism is given by a pair of matrices of $q$-series with integer coefficients, which are determined explicitly by the fundamental solutions of a pair of linear $q$-difference equations. We further conjecture that for a hyperbolic knot, a distinguished entry of those matrices equals to the Dimofte-Gaiotto-Gukov 3D-index, and thus is given by a counting of BPS states. We illustrate our conjectures explicitly by matching theoretically and numerically computed integers for the cases of the $4_1$ and the $5_2$ knots.