Khovanov homology and equivariant surfaces

TL;DR

This paper develops a refined homology theory for involutive links, introducing new invariants that reveal significant differences between equivariant and isotopy-equivariant slice genera, advancing understanding of equivariant surfaces.

Contribution

It extends Bar-Natan homology to involutive links, creating new invariants and bounds for equivariant cobordisms, and demonstrates the potential for large differences between slice genera.

Findings

New invariants for involutive links

Bounds for equivariant cobordism genus

Difference between slice genera can be arbitrarily large

Abstract

We introduce a refinement of Bar-Natan homology for involutive links, extending the work of Lobb-Watson and Sano. We construct a new suite of numerical invariants and derive bounds for the genus of equivariant cobordisms between strongly invertible knots. Our invariants show that the difference between the equivariant slice genus and isotopy-equivariant slice genus can be arbitrarily large, whereas previously these were not known to differ.