A two-variable series for knot complements

Abstract

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. Furthermore, there is a formula that relates $F_K(x,q)$ to the invariants ${\hat{Z}}_a(q)$ for Dehn surgeries on the knot. We provide explicit calculations of $F_K(x,q)$ in the case of knots given by negative definite plumbings with an unframed vertex, such as torus knots. We also find numerically the first terms in the series for the figure-eight knot, up to any desired order, and use this to understand ${\hat{Z}}_a(q)$ for some hyperbolic 3-manifolds.