Abstract:

I will discuss a new invariant of negative definite plumbed 3-manifolds called a weighted graded root. It unifies and extends two theories with quite different origins and structures. The first is lattice cohomology, due to Némethi, which is motivated by normal surface singularities and is isomorphic to Heegaard Floer homology for a large class of plumbings. The second theory is the Z-hat q-series of Gukov-Pei-Putrov-Vafa, a power series which conjecturally recovers SU(2) quantum invariants at roots of unity and satisfies remarkable modularity properties. I will explain lattice cohomology, Z-hat, and our unification of these theories. I will also discuss some key features of the weighted graded root: it leads to a 2-variable refinement of Z-hat, and, unlike both lattice cohomology and Z-hat, it is sensitive to spin^c conjugation. This is joint work with Peter Johnson and Slava Krushkal.