# A two-variable series for knot complements

## TL;DR

This paper introduces a two-variable series invariant for knot complements, derived from the asymptotic expansion of the colored Jones polynomial, and explores its relation to 3-manifold invariants and quantum A-polynomials.

## Contribution

It defines a new two-variable series $F_K(x,q)$ for knot complements, connecting quantum invariants, resurgence, and the quantum A-polynomial, with explicit calculations for specific knots.

## Key findings

- Explicit formulas for $F_K(x,q)$ for certain plumbed knots.
- Numerical computations of $F_K(x,q)$ for the figure-eight knot.
- Insights into $ ext{Z}_a(q)$ invariants for hyperbolic 3-manifolds.

## 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.