Bigrading the symplectic Khovanov cohomology

TL;DR

This paper introduces a second grading on symplectic Khovanov cohomology, aligning it with the Jones grading of Khovanov homology, and refines existing isomorphisms between these theories.

Contribution

It constructs a well-defined bigrading on symplectic Khovanov cohomology and demonstrates its compatibility with Khovanov homology's Jones grading, refining prior isomorphisms.

Findings

Defined a relative second grading via holomorphic disc counting.

Refined the Abouzaid-Smith isomorphism to a bigraded isomorphism.

Established an exact triangle analogous to the skein relation.

Abstract

We construct a well-defined relative second grading on symplectic Khovanov cohomology from holomorphic disc counting. We show that it 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-Smith can be refined as an isomorphism between bigraded cohomology theories. We prove it by constructing an exact triangle of symplectic Khovanov cohomology that behaves similarly to the unoriented skein exact triangle for Khovanov homology. We use a version of symplectic Khovanov cohomology defined for bridge diagrams and obtain an absolute homological grading in this construction.