  Using resurgent analysis we offer a novel mathematical perspective on a
curious bijection (duality) that has many potential applications ranging from
the theory of vertex algebras to the physics of SCFTs in various dimensions, to
q-series invariants in low-dimensional topology that arise, e.g. in Vafa-Witten
theory and in non-perturbative completion of complex Chern-Simons theory. In
particular, we introduce explicit numerical algorithms that efficiently
implement this bijection. This bijection is founded on preservation of
relations, a fundamental property of resurgent functions. Using resurgent
analysis we find new structures and patterns in complex Chern-Simons theory on
closed hyperbolic 3-manifolds obtained by surgeries on hyperbolic twist knots.
The Borel plane exhibits several intriguing hints of a new form of
integrability. An important role in this analysis is played by the twisted
Alexander polynomials and the adjoint Reidemeister torsion, which help us
determine the Stokes data. The method of singularity elimination enables
extraction of geometric data even for very distant Borel singularities, leading
to detailed non-perturbative information from perturbative data. We also
introduce a new double-scaling limit to probe 0-surgeries from the limiting
$r\to\infty$ behavior of 1/r surgeries, and apply it to the family of
hyperbolic twist knots.
