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Bigrading the symplectic Khovanov cohomology

Zhechi Cheng

Algebraic & Geometric Topology 23 (2023) 4057–4086
Abstract

We construct a well-defined relative second grading on symplectic Khovanov cohomology from holomorphic disc counting. We use a version of symplectic Khovanov cohomology defined for bridge diagrams rather than braids. We show that our second grading recovers the Jones grading of Khovanov homology up to an overall grading shift over any characteristic-zero field, through proving that the isomorphism of Abouzaid and Smith can be refined as an isomorphism between bigraded cohomology theories. The central idea of the proof is to construct an exact triangle for symplectic Khovanov cohomology that behaves similarly to the unoriented skein exact triangle for Khovanov homology.

Keywords
Khovanov homology, Fukaya category, symplectic Khovanov cohomology, Hilbert scheme
Mathematical Subject Classification
Primary: 53D40, 57K18, 57K10, 57R58
References
Publication
Received: 1 February 2021
Revised: 16 May 2022
Accepted: 8 June 2022
Published: 23 November 2023
Authors
Zhechi Cheng
School of Mathematics and Statistics
Wuhan University
Wuhan, Hubei
China

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