Summability for State Integrals of hyperbolic knots

TL;DR

This paper proves that certain series related to hyperbolic knots are resurgent and Borel summable, providing an algorithm to explicitly compute their resummation and analyze their Stokes phenomena.

Contribution

It confirms conjectures about the resurgent nature of series for specific hyperbolic knots and introduces an explicit algorithm for their Borel-Laplace resummation.

Findings

Series for knots 4_1 and 5_2 are resurgent and Borel summable.

An explicit algorithm for computing Borel-Laplace resummation is provided.

Complete description of the resurgent structure and Stokes constants for these knots.

Abstract

We prove conjectures of Garoufalidis-Gu-Mari\~no that perturbative series associated with the hyperbolic knots $4_1$ and $5_2$ are resurgent and Borel summable. In the process, we give an algorithm that can be used to explicitly compute the Borel-Laplace resummation as a combination of state integrals of Andersen-Kashaev. This gives a complete description of the resurgent structure in these examples and allows for explicit computations of Stokes constants.