The flip map and involutions on Khovanov homology

TL;DR

This paper investigates the involution induced by flip symmetry on Khovanov homology, confirming a folklore conjecture and exploring its behavior on unlinks and specific knot diagrams.

Contribution

It proves that the flip involution is trivial over  and shows how symmetries on certain knot diagrams induce the same involution on Khovanov homology.

Findings

The flip involution is the identity over .

Symmetries on specific knot diagrams induce the same involution.

The techniques apply to the half sweep-around map.

Abstract

The flip symmetry on knot diagrams induces an involution on Khovanov homology. We prove that this involution is determined by its behavior on unlinks; in particular, it is the identity map when working over $\mathbb{F}_2$. This confirms a folklore conjecture on the triviality of the Viro flip map. As a corollary, we prove that the symmetries on the transvergent and intravergent diagrams of a strongly invertible knot induce the same involution on Khovanov homology. We also apply similar techniques to study the half sweep-around map.