# The resurgent structure of quantum knot invariants

Stavros Garoufalidis, Jie Gu, Marcos Marino

Stavros Garoufalidis, Jie Gu, Marcos Marino

**20/7/20**Published in : arXiv:2007.10190

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.