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# Quantum Topology

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**Volume 12, Issue 1, 2021, pp. 1–109**

**DOI: 10.4171/QT/145**

Published online: 2021-03-15

A two-variable series for knot complements

Sergei Gukov^{[1]}and Ciprian Manolescu

^{[2]}(1) California Institute of Technology, Pasadena, USA and Max Planck Institute, Bonn, Germany

(2) University of California, Los Angeles, USA

The physical 3d $\mathcal N = 2$ theory $T[Y]$ was previously used to predict the existence of some $3$-manifold invariants $\widehat{Z}_{a}(q)$ 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 Wittenâ€“Reshetikhinâ€“Turaev invariants. In this paper we discuss how, for complements of knots in $S^3$, the analogue of the invariants $\widehat{Z}_{a}(q)$ 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 $\widehat{Z}_{a}(q)$ 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 $\widehat{Z}_a(q)$ for some hyperbolic 3-manifolds.

*Keywords: *WRT invariants, BPS states, Dehn surgery, resurgence, colored Jones polynomial

Gukov Sergei, Manolescu Ciprian: A two-variable series for knot complements. *Quantum Topol.* 12 (2021), 1-109. doi: 10.4171/QT/145