Knot Floer homology and Khovanov homology are homological knot invariants that are defined using very different methods — the former is a Lagrangian Floer homology, while the latter has roots in representation theory. Despite these differences, the two theories contain a great deal of the same information and were conjectured by Rasmussen to be related by a spectral sequence. This conjecture was recently proved by Dowlin, however, his proof is not computationally effective. In this talk we will sketch a local framework for proving this conjecture. To do that, we will describe an algebraic/combinatorial glueable braid invariant which using a specific closing up operation results in a knot invariant related by a spectral sequence to Khovanov homology. Moreover, it is chain homotopic to Ozsvath–Szabo's braid invariants which using their closing up operation recovers knot Floer homology. If time permits we will compare the two closing up operations. This is joint work with Nathan Dowlin.
This talk was part of the conference "Perspectives on quantum link homology theories", see https://cbz20.raspberryip.com/Perspectives-2021