Computations of the Lipshitz-Sarkar Steenrod Square on Khovanov Homology

TL;DR

This paper computes the Lipshitz-Sarkar Steenrod square on Khovanov homology, demonstrating how it distinguishes links with identical Khovanov homology but different homotopy types, advancing the understanding of link invariants.

Contribution

It provides explicit computations of the Steenrod square on Khovanov homology and shows its effectiveness in differentiating links with identical homology but distinct homotopy types.

Findings

Examples of links with same Khovanov homology but different homotopy types.

Explicit calculations of the Steenrod square on various links.

Demonstration of the Steenrod square's discriminative power.

Abstract

Lipshitz and Sarkar recently introduced a space-level refinement of Khovanov homology. This refinement induces a Steenrod square operation $\Sq^2$ on Khovanov homology which they describe explicitly. This paper presents some computations of $\Sq^2$. In particular, we give examples of links with identical integral Khovanov homology but with distinct Khovanov homotopy types.