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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 Witten–Reshetikhin–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