Quivers for 3-manifolds: the correspondence, BPS states, and 3d $$ \mathcal{N} $$ = 2 theories

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 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 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.