  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.
