  The partition function of complex Chern-Simons theory on a 3-manifold with
torus boundary reduces to a finite dimensional state-integral which is a
holomorphic function of a complexified Planck's constant $\tau$ in the complex
cut plane and an entire function of a complex parameter $u$. This gives rise to
a vector of factorially divergent perturbative formal power series whose Stokes
rays form a peacock-like pattern in the complex plane.
  We conjecture that these perturbative series are resurgent, their
trans-series involve two non-perturbative variables, their Stokes automorphism
satisfies a unique factorization property and that it is given explicitly in
terms of a fundamental matrix solution to a (dual) linear $q$-difference
equation. We further conjecture that entries of the Stokes automorphism matrix
are the 3D-indices of Dimofte-Gaiotto-Gukov. We provide proofs of our
statements regarding the $q$-difference equations and their properties of their
fundamental solutions and illustrate our conjectures regarding the Stokes
matrices with numerical calculations for the two simplest hyperbolic $4_1$ and
$5_2$ knots.
