Knot invariants arising from homological operations on Khovanov homology

TL;DR

This paper introduces a new algebraic structure of homological operations on Khovanov homology, enhancing knot invariants and distinguishing knots beyond traditional homology methods.

Contribution

It constructs an algebra of homological operations on Khovanov homology and lifts it to integral coefficients, proposing conjectures about its infinitude and demonstrating its finer distinguishing power.

Findings

Constructed an algebra generated by Bockstein operations on Khovanov homology.

Provided evidence supporting the conjecture that the algebra is infinite.

Identified knots with identical even and odd Khovanov homology but different homological operations.

Abstract

We construct an algebra of non-trivial homological operations on Khovanov homology with coefficients in $\mathbb Z_2$ generated by two Bockstein operations. We use the unified Khovanov homology theory developed by the first author to lift this algebra to integral Khovanov homology. We conjecture that these two algebras are infinite and present evidence in support of our conjectures. Finally, we list examples of knots that have the same even and odd Khovanov homology, but different actions of these homological operations. This confirms that the unified theory is a finer knot invariant than the even and odd Khovanov homology combined. The case of reduced Khovanov homology is also considered.