Immersed curves in Khovanov homology

Abstract

We give a geometric interpretation of Bar-Natan's universal invariant for the class of tangles in the 3-ball with four ends: we associate with such 4-ended tangles $T$ multicurves $\widetilde{\mathrm{BN}}(T)$, that is, collections of immersed curves with local systems in the 4-punctured sphere. These multicurves are tangle invariants up to homotopy of the underlying curves and equivalence of the local systems. They satisfy a gluing theorem which recovers the reduced Bar-Natan homology of links in terms of wrapped Lagrangian Floer theory. Furthermore, we use $\widetilde{\mathrm{BN}}(T)$ to define two immersed curve invariants $\widetilde{\mathrm{Kh}}(T)$ and $\mathrm{Kh}(T)$, which satisfy similar gluing theorems that recover reduced and unreduced Khovanov homology of links, respectively. As a first application, we prove that Conway mutation preserves reduced Bar-Natan homology over the field with two elements and Rasmussen's $s$-invariant over any field. As a second application, we give a geometric interpretation of Rozansky's categorification of the two-stranded Jones-Wenzl projector. This allows us to define a module structure on reduced Bar-Natan and Khovanov homologies of infinitely twisted knots, generalizing a result by Benheddi.

arXiv comment95 pages, 62 figures. Uses TikZ and PSTricks, also colour. Version 2: minor edits throughout, including corrected discussion around some of the background material in Section 3

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• The Fukaya category of the pillowcase, traceless character varieties, and Khovanov Cohomology
• $U_q(sl(2))$-quantum invariants from an intersection of two Lagrangians in a symmetric power of a surface