Immersed curves in Khovanov homology
Immersed curves in Khovanov homology
Consider a 2-sphere S intersecting a knot K in 4Ìýpoints. This defines decomposition of a knot into two 4-ended tangles.ÌýWe will show that Khovanov homology Kh(K), and its deformation due toÌýBar-Natan, are isomorphic to wrapped Lagrangian Floer homology of a pair ofÌýspecifically constructed immersed curves on the dividing 4-puncturedÌýsphere S.ÌýThis result is analogous to immersed curves description ofÌýbordered Heegaard Floer homology and knot Floer homology.ÌýThe key stepÌýwill be constructing a tangle invariant in the form of a chain complexÌýover a certain algebra B (deformation of Khovanov's arc algebra), andÌýshowing that algebra B embeds into the wrapped FukayaÌýcategory of the 4-punctured sphere. As an application, we willÌýprove that Conway mutation preserves Rasmussen's s-invariant of knots.ÌýThis is joint work with Liam Watson and Claudius Zibrowius.