Khovanov homology from Floer cohomology

Abouzaid, Mohammed¹ ; Smith, Ivan²

¹ Department of Mathematics, Columbia University, 2990 Broadway, New York, New York 10027
² Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, England

Journal of the American Mathematical Society, Tome 32 (2019) no. 1, pp. 1-79

Résumé

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.

Détail

DOI : 10.1090/jams/902

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