Two second Steenrod squares for odd Khovanov homology

TL;DR

This paper introduces a method to compute the second Steenrod square in odd Khovanov homology, providing a link invariant that refines the Rasmussen s-invariant with potential connections to recent stable homotopy type constructions.

Contribution

It proposes a new computational approach for the second Steenrod square in odd Khovanov homology, refining the s-invariant and linking to recent homotopy type developments.

Findings

The construction is a link invariant.

It refines the Rasmussen s-invariant.

Potential relation to Sarkar-Scaduto-Stoffregen's Steenrod square.

Abstract

Recently, Sarkar-Scaduto-Stoffregen constructed a stable homotopy type for odd Khovanov homology, hence obtaining an action of the Steenrod algebra on Khovanov homology with $\mathbb{Z}/2\mathbb{Z}$ coefficients. Motivated by their construction we propose a way to compute the second Steenrod square. Our construction is not unique, but we can show it to be a link invariant which gives rise to a refinement of the Rasmussen $s$-invariant with $\mathbb{Z}/2\mathbb{Z}$ coefficients. We expect it to be related to the second Steenrod square arising from the Sarkar-Scaduto-Stoffregen construction.