  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.
