Khovanov homology via immersed curves

Khovanov homology via immersed curves

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Artem Kotelskiy, Indiana University

Zoom link:  https://princeton.zoom.us/j/453512481?pwd=OHZ5TUJvK2trVVlUVmJLZkhIRHFDUT09

We will describe a novel Floer-theoretic approach to Khovanov homology, where the topological input is a Conway two-sphere S intersecting a knot K in 4 points. The geometric outcome is that Khovanov homology Kh(K), and its deformation BN(K) due to Bar-Natan, are isomorphic to the wrapped Floer homology of a pair of specifically constructed immersed curves on the dividing 4-punctured sphere S. We will also describe a connection of our framework to the homological mirror symmetry statement for the three-punctured sphere, via the matrix factorization framework of Khovanov-Rozansky. This will restrict the geometry of curve invariants, making it possible to apply immersed curves to tangle replacement questions. In particular, we will discuss mutation invariance of Kh(K;Q) and the Generalised Cosmetic Crossing Conjecture.

This is joint work with Liam Watson and Claudius Zibrowius.