A two-variable series for knot complements

Understand

The physical 3d $\mathcal{N}=2$ theory T[Y] was previously used to predict the existence of some 3-manifold invariants $\hat{Z}_{a}(q)$ that take the form of power series with integer coefficients, converging in the unit disk.

• Their radial limits at the roots of unity should recover the Witten-Reshetikhin-Turaev invariants.
• In this paper we discuss how, for complements of knots in $S^3$, the analogue of the invariants $\hat{Z}_{a}(q)$ should be a two-variable series $F_K(x,q)$ obtained by parametric resurgence from the asymptotic expansion of the colored Jones polynomial.
• The terms in this series should satisfy a recurrence given by the quantum A-polynomial.