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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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Khovanov homology from Floer cohomology
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by Mohammed Abouzaid and Ivan Smith
J. Amer. Math. Soc. 32 (2019), 1-79
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).


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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Bibliographic 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
  • © Copyright 2018 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 32 (2019), 1-79
  • MSC (2010): Primary 53D40; Secondary 53D37, 57M25
  • DOI:
  • MathSciNet review: 3867999