Appears in collection : Mathematics Inspired by Physics

Given a hyperbolic knot, the Andersen-Kashaev state integrals are convergent integrals built from certain triangulations of the knot complement. Their asymptotic expansion is a perturbative topological invariant of the knot, conjectured to be resurgent and Borel summable by Garoute falidis, Gu, and Mariño. In this talk, I will present the main ideas of the proofs of these conjectures, based on a joint project with Wheeler (arXiv:2410.20973) and our ongoing work together with J. E. Andersen and M. Kontsevich.