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Khovanov homology from Floer cohomology

Authors: Mohammed Abouzaid and Ivan Smith
Journal: J. Amer. Math. Soc. 32 (2019), 1-79
MSC (2010): Primary 53D40; Secondary 53D37, 57M25
Published electronically: July 27, 2018
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).
MathSciNet review: 3867999
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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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Additional Information

Mohammed Abouzaid
Affiliation: Department of Mathematics, Columbia University, 2990 Broadway, New York, New York 10027
MR Author ID: 734175

Ivan Smith
Affiliation: Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, England
MR Author ID: 650668

Received by editor(s): April 6, 2015
Received by editor(s) in revised form: December 31, 2017
Published electronically: July 27, 2018
Additional Notes: The first author was partially supported by NSF grants DMS-1308179, DMS-1609148, and DMS-1564172, and by the Simons Foundation through its “Homological Mirror Symmetry” collaboration grant
The second author is partially supported by a fellowship from EPSRC
Article copyright: © Copyright 2018 American Mathematical Society