Title: Khovanov homology and Bar-Natan's deformation via immersed curves in the 4-punctured sphere.

Date: 10/04/2018

Time: 2:00 PM - 2:50 PM

Place: C304 Wells Hall

We will describe a geometric interpretation of Khovanov homology and its deformation due to Bar-Natan as Lagrangian Floer homology of two immersed curves in the 4-punctured 2-sphere S^2 \ 4pt. We will first start with a certain cobordism theoretic algebra H, where elements are all cobordisms between two trivial tangles )( and = up to certain relations. The central point then will be the observation that this algebra is isomorphic to an algebra B = Fuk(a0, a1), whose elements are generators of wrapped Lagrangian Floer complexes between two arcs a0 and a1 inside S^2 \ 4pt. The results will follow because D structures over H give Khovanov/Bar-Natan invariants for 4-ended tangles, and D structures over B give curves in S^2 \ 4pt (due to [Haiden, Katzarkov, Kontsevich]).
The construction is originally inspired by a result of [Hedden, Herald, Hogancamp, Kirk], which embeds 4-ended reduced Khovanov arc algebra (or, equivalently, Bar-Natan dotted cobordism algebra) into the Fukaya category of the 4-punctured sphere. This is joint work with Liam Watson and Claudius Zibrowius.