  The purpose of the paper is to introduce some conjectures regarding the
analytic continuation and the arithmetic properties of quantum invariants of
knotted objects. More precisely, we package the perturbative and
nonperturbative invariants of knots and 3-manifolds into two power series of
type P and NP, convergent in a neighborhood of zero, and we postulate their
arithmetic resurgence. By the latter term, we mean analytic continuation as a
multivalued analytic function in the complex numbers minus a discrete set of
points, with restricted singularities, local and global monodromy. We point out
some key features of arithmetic resurgence in connection to various problems of
asymptotic expansions of exact and perturbative Chern-Simons theory with
compact or complex gauge group. Finally, we discuss theoretical and
experimental evidence for our conjecture.

Geometric Topology;Number Theory