A Steenrod Square on Khovanov Homology

TL;DR

This paper introduces a Steenrod square operation on Khovanov homology, computes it for links up to 11 crossings, and uses this to determine the stable homotopy type of the associated space-level invariant.

Contribution

It defines and computes the Sq^2 operation on Khovanov homology, advancing the understanding of the space-level invariant X(L) and its algebraic structure.

Findings

Computed Sq^2 for all links up to 11 crossings

Determined the stable homotopy type of X(L) for these links

Enhanced the algebraic understanding of Khovanov homology operations

Abstract

In a previous paper, we defined a space-level version X(L) of Khovanov homology. This induces an action of the Steenrod algebra on Khovanov homology. In this paper, we describe the first interesting operation, Sq^2:Kh^{i,j}(L) -> Kh^{i+2,j}(L). We compute this operation for all links up to 11 crossings; this, in turn, determines the stable homotopy type of X(L) for all such links.