Papers › Immersed curves in Khovanov homology
Immersed curves in Khovanov homology
Artem Kotelskiy, Liam Watson, Claudius Zibrowius
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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 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 BN(T) to define two immersed curve invariants Kh(T) and 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.
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