Authors: Ekholm, Tobias
Gruen, Angus
Gukov, Sergei
Kucharski, Piotr
Park, Sunghyuk
Stošić, Marko 
Sułkowski, Piotr
Affiliations: Mathematics 
Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Branches, quivers, and ideals for knot complements
Journal: Journal of Geometry and Physics
Volume: 177
First page: 104520
Issue Date: 1-Jul-2022
Rank: ~M22
ISSN: 0393-0440
DOI: 10.1016/j.geomphys.2022.104520
We generalize the FK invariant, i.e. Zˆ 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 FK invariant to any branch of the A-polynomial of K and we work out explicit expressions for several simple knots. We show that these FK 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].
Keywords: A polynomial | Open curve counts | Quantum invariants; High Energy Physics - Theory; High Energy Physics - Theory; Mathematics - Geometric Topology; Mathematics - Quantum Algebra; Mathematics - Symplectic Geometry
Publisher: Elsevier

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