Khovanov homology from Floer cohomology

Abouzaid, Mohammed; Smith, Ivan

doi:10.1090/jams/902

Khovanov homology from Floer cohomology
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by Mohammed Abouzaid and Ivan Smith

J. Amer. Math. Soc. 32 (2019), 1-79

DOI: https://doi.org/10.1090/jams/902

Published electronically: July 27, 2018

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Previous version: Original version posted July 27, 2018
Corrected version: Current version corrects publisher's error which introduced a spurious ``i'' before the overcrossing symbols in equations (7.7) and (7.9) and on the first line of text following equation (7.9).

Abstract:

This paper realises the Khovanov homology of a link in $S^3$ 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 $\mathbf {k}$ of characteristic zero. Here we prove the symplectic cup and cap bimodules, which relate different symplectic arc algebras, are themselves formal over $\mathbf {k}$, and we construct a long exact triangle for symplectic Khovanov cohomology. We then prove the symplectic and combinatorial arc algebras are isomorphic over $\mathbb {Z}$ in a manner compatible with the cup bimodules. It follows that Khovanov cohomology and symplectic Khovanov cohomology co-incide in characteristic zero.

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