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Branches, quivers, and ideals for knot complements

Ekholm, Tobias and Gruen, Angus and Gukov, Sergei and Kucharski, Piotr and Park, Sunghyuk and Stošić, Marko and Sułkowski, Piotr (2022) Branches, quivers, and ideals for knot complements. Journal of Geometry and Physics, 177 . Art. No. 104520. ISSN 0393-0440. doi:10.1016/j.geomphys.2022.104520.

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We generalize the F_K invariant, i.e. Ẑ for the complement of a knot K in the 3-sphere, the knots-quivers correspondence, and A-polynomials of knots, and find several interconnections between them. We associate an F_K invariant to any branch of the A-polynomial of K and we work out explicit expressions for several simple knots. We show that these F_K invariants can be written in the form of a quiver generating series, in analogy with the knots-quivers correspondence. We discuss various methods to obtain such quiver representations, among others using R-matrices. We generalize the quantum a-deformed A-polynomial to an ideal that contains the recursion relation in the group rank, i.e. in the parameter a, and describe its classical limit in terms of the Coulomb branch of a 3d-5d theory. We also provide t-deformed versions. Furthermore, we study how the quiver formulation for closed 3-manifolds obtained by surgery leads to the superpotential of 3d N = 2 theory T[M₃] and to the data of the associated modular tensor category MTC[M₃].

Item Type:Article
Related URLs:
URLURL TypeDescription
Gruen, Angus0000-0003-0284-009X
Kucharski, Piotr0000-0002-9599-5658
Park, Sunghyuk0000-0002-6132-0871
Sułkowski, Piotr0000-0002-6176-6240
Additional Information:© 2022 Elsevier. Received 21 December 2021, Accepted 25 March 2022, Available online 1 April 2022, Version of Record 12 April 2022. P.K. was supported by the Polish Ministry of Education and Science through its programme Mobility Plus (decision number 1667/MOB/V/2017/0) and by NWO vidi grant (number 016.Vidi.189.182). S.P. was partially supported by junior fellowship at Institut Mittag-Leffler and by Kwanjeong Educational Foundation. The work of M.S. was supported by the Portuguese Fundação para a Ciência e a Tecnologia (FCT) through the grant ‘Higher Structures and Applications’, no. PTDC/MAT-PUR/31089/2017, and FCT Exploratory Grant IF/0998/2015, and by the Ministry of Education, Science and Technological Development of the Republic of Serbia through Mathematical Institute SANU. In the final stages, his work was supported by the Science Fund of the Republic of Serbia, Grant No. 7749891, Graphical Languages – GWORDS. The work of P.S. was supported by the TEAM programme of the Foundation for Polish Science co-financed by the European Union under the European Regional Development Fund (POIR.04.04.00-00-5C55/17-00). T.E. is supported by the Knut and Alice Wallenberg Foundation as a Wallenberg scholar KAW2020.0307 and by the Swedish Research Council VR2020-04535. The work of S.G. is 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. NSF DMS 1664227.
Funding AgencyGrant Number
Ministry of Education and Science (Poland)1667/MOB/V/2017/0
Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO)016.Vidi.189.182
Institut Mittag-LefflerUNSPECIFIED
Kwanjeong Educational FoundationUNSPECIFIED
Fundação para a Ciência e a Tecnologia (FCT)PTDC/MAT-PUR/31089/2017
Fundação para a Ciência e a Tecnologia (FCT)IF/0998/2015
Ministry of Education, Science and Technological Development (Serbia)UNSPECIFIED
Science Fund (Serbia)7749891
Foundation for Polish ScienceUNSPECIFIED
European Regional Development FundPOIR.04.04.00-00-5C55/17-00
Knut and Alice Wallenberg FoundationKAW2020.0307
Swedish Research CouncilVR2020-04535
Department of Energy (DOE)DE-SC0011632
Subject Keywords:Quantum invariants; A polynomial; Open curve counts
Record Number:CaltechAUTHORS:20220412-15486000
Persistent URL:
Official Citation:Tobias Ekholm, Angus Gruen, Sergei Gukov, Piotr Kucharski, Sunghyuk Park, Marko Stošić, Piotr Sułkowski, Branches, quivers, and ideals for knot complements, Journal of Geometry and Physics, Volume 177, 2022, 104520, ISSN 0393-0440, (
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:114255
Deposited By: George Porter
Deposited On:13 Apr 2022 15:23
Last Modified:13 Apr 2022 15:23

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