Khovanov homology from Floer cohomology
By M. Abouzaid, I. Smith. Journal of The American Mathematical Society, 2015
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This paper realises the Khovanov homology of a link in S 3 S^3 as a Lagrangian Floer cohomology group, establishing a conjecture of Seidel and the second author. The starting point is the previously established formality theorem for the symplectic arc algebra over a field k \mathbf {k} of characteristic zero. Here we prove the symplectic cup and cap bimodules, which relate different symplectic arc algebras, are themselves formal over k \mathbf {k} , and we construct a long exact triangle for symplectic Khovanov cohomology. We then prove the symplectic and combinatorial arc algebras are isomorphic over Z \mathbb {Z} in a manner compatible with the cup bimodules. It follows that Khovanov cohomology and symplectic Khovanov cohomology co-incide in characteristic zero.
Subfields: Geometry and Topology, Mathematical Physics, Neurology
View Paper: https://www.semanticscholar.org/paper/60544a9caa161d1d6245f6349e7a6973a5ce5e44
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- Bounds on spectral norms and barcodes (2018, background): “ the modules are homologically unital, it can be given, in homology, by the operator of multiplication by a suitable homology class. This class is called a continuation element in the literature (cf. [2, 4, 48, 66]). Finally, considering the chain level and ltrations, it turns out benecial to replac”