Branches, quivers, and ideals for knot complements
arXiv:2110.13768 · doi:10.1016/j.geomphys.2022.104520
Abstract
We generalize the invariant, i.e. for the complement of a knot in the 3-sphere, the knots-quivers correspondence, and -polynomials of knots, and find several interconnections between them. We associate an invariant to any branch of the -polynomial of and we work out explicit expressions for several simple knots. We show that these 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 -matrices. We generalize the quantum -deformed -polynomial to an ideal that contains the recursion relation in the group rank, i.e. in the parameter , and describe its classical limit in terms of the Coulomb branch of a 3d-5d theory. We also provide -deformed versions. Furthermore, we study how the quiver formulation for closed 3-manifolds obtained by surgery leads to the superpotential of 3d theory and to the data of the associated modular tensor category .
arXiv comment99 pages, 13 figures
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