Georgia Institute of Technology College of Sciences

We will describe a geometric interpretation of Khovanov homology as Lagrangian Floer homology of two immersed curves in the 4-punctured 2-dimensional sphere. The main ingredient is a construction which associates an immersed curve to a 4-ended tangle. This curve is a geometric way to represent Khovanov (or Bar-Natan) invariant for a tangle. We will show that for a rational tangle the curve coincides with the representation variety of the tangle complement. The construction is 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. The main tool we will use is a category of peculiar modules, introduced by Zibrowius, which is a model for the Fukaya category of a 2-sphere with 4 discs removed. This is joint work with Claudius Zibrowius and Liam Watson.