# Immersed curves in Khovanov homology

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Artem Kotelskiy, Indiana University
Fine Hall 314

Consider a 2-sphere S intersecting a knot K in 4 points. This defines decomposition of a knot into two 4-ended tangles. We will show that Khovanov homology Kh(K), and its deformation due to Bar-Natan, are isomorphic to wrapped Lagrangian Floer homology of a pair of specifically constructed immersed curves on the dividing 4-punctured sphere S. This result is analogous to immersed curves description of bordered Heegaard Floer homology and knot Floer homology. The key step will be constructing a tangle invariant in the form of a chain complex over a certain algebra B (deformation of Khovanov's arc algebra), and showing that algebra B embeds into the wrapped Fukaya category of the 4-punctured sphere. As an application, we will prove that Conway mutation preserves Rasmussen's s-invariant of knots. This is joint work with Liam Watson and Claudius Zibrowius.