Abstract
We introduce and explore the relation between quivers and 3-manifolds with the topology of the knot complement. This idea can be viewed as an adaptation of the knots-quivers correspondence to Gukov-Manolescu invariants of knot complements (also known as FK or hat{Z} ). Apart from assigning quivers to complements of T(2,2p+1) torus knots, we study the physical interpretation in terms of the BPS spectrum and general structure of 3d mathcal{N} = 2 theories associated to both sides of the correspondence. We also make a step towards categorification by proposing a t-deformation of all objects mentioned above.
Highlights
Torus knots and twist knots in [2]
We introduce and explore the relation between quivers and 3-manifolds with the topology of the knot complement
Apart from assigning quivers to complements of T (2,2p+1) torus knots, we study the physical interpretation in terms of the BPS spectrum and general structure of 3d N = 2 theories associated to both sides of the correspondence
Summary
If K ⊂ S3 is a knot, its HOMFLY-PT polynomial PK(a, q) [49, 50] is a topological invariant which can be calculated via the skein relation. HOMFLY-PT polynomials PK,R(a, q) are similar polynomial knot invariants depending on a representation R of the Lie algebra u(N ). In this setting, the original HOMFLY-PT corresponds to the fundamental representation. In the context of the knots-quivers correspondence, we are interested in the HOMFLY-. HOMFLY-PT polynomials satisfy recurrence relations encoded in the quantum adeformed A-polynomials [14,15,16,17,18]: A(μ, λ, a, q)PK,r(a, q) = 0,. LMOV invariants can be extracted from the A-polynomials, see [52, 53]
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