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A two-variable series for knot complements

Gukov, Sergei and Manolescu, Ciprian (2021) A two-variable series for knot complements. Quantum Topology, 12 (1). pp. 1-109. ISSN 1663-487X. doi:10.4171/QT/145.

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The physical 3d N=2 theory T[Y] was previously used to predict the existence of some 3-manifold invariants Za(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 S³, the analogue of the invariants Za(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 Za(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 Za(q) for some hyperbolic 3-manifolds.

Item Type:Article
Related URLs:
URLURL TypeDescription Paper
Gukov, Sergei0000-0002-9486-1762
Additional Information:© 2021 European Mathematical Society. Published by EMS Press. This work is licensed under a CC BY 4.0 license. Received June 7, 2019. Published online: 2021-03-15. Sergei Gukovwas supported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award No. DE-SC0011632, and by the National Science Foundation under Grant No. DMS 1664240. Ciprian Manolescu was supported by the National Science Foundation under Grant No. DMS-1708320.
Funding AgencyGrant Number
Department of Energy (DOE)DE-SC0011632
Subject Keywords:WRT invariants, BPS states, Dehn surgery, resurgence, colored Jones polynomial
Issue or Number:1
Classification Code:Mathematics Subject Classification (2020): Primary: 57K16, Secondary: 57K14, 57R56
Record Number:CaltechAUTHORS:20210413-133913817
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Official Citation:A two-variable series for knot complements. Gukov Sergei, Manolescu Ciprian: A two-variable series for knot complements. Quantum Topol. 12 (2021), 1-109. doi: 10.4171/QT/145
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:108717
Deposited By: Tony Diaz
Deposited On:13 Apr 2021 21:51
Last Modified:19 Apr 2021 22:40

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