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Lattice cohomology and q-series invariants of 3-manifolds

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Ross Akhmechet (Columbia)

Abstract: I will discuss joint work with Peter Johnson and Slava Krushkal which introduces an invariant of negative definite plumbed 3-manifolds that unifies and extends two theories with quite different origins. The first is lattice cohomology, due to NĂ©methi, whose degree zero part is described by a certain graph and is isomorphic to Heegaard Floer homology for a large subclass of such 3-manifolds. The second theory is the Z-hat q-series of Gukov-Pei-Putrov-Vafa, a power series which recovers SU(2) quantum invariants at roots of unity. I will explain lattice cohomology, Z-hat, and our unification of these theories. I will also mention work in preparation with Peter Johnson and Sunghyuk Park on extending the construction to knot complements.