On a triply graded Khovanov homology

TL;DR

This paper introduces a triply graded version of Khovanov homology, which, while not strengthening the invariant, reveals multiplicative properties and module structures related to link operations.

Contribution

It extends Khovanov homology to a triply graded theory and demonstrates its applications to multiplicativity and module actions on link invariants.

Findings

Odd Khovanov homology is multiplicative under disjoint unions and connected sums.

The triply graded structure does not enhance the invariant's strength.

Sliding a basepoint through a crossing affects the module structure on homology.

Abstract

Cobordisms are naturally bigraded and we show that this grading extends to Khovanov homology, making it a triply graded theory. Although the new grading does not make the homology a stronger invariant, it can be used to show that odd Khovanov homology is multiplicative with respect to disjoint unions and connected sums of links; same results hold for the generalized Khovanov homology defined by the author in his previous work. We also examine the module structure on both odd and even Khovanov homology, in particular computing the effect of sliding a basepoint through a crossing on the integral homology.