Bigrading the symplectic Khovanov cohomology

Cheng, Zhechi

doi:10.2140/agt.2023.23.4057

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Bigrading the symplectic Khovanov cohomology

Zhechi Cheng

Algebraic & Geometric Topology 23 (2023) 4057–4086

DOI: 10.2140/agt.2023.23.4057

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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Volume 23, issue 9 (2023)