A Steenrod square on Khovanov homology and a cup-i product

TL;DR

This paper proves that Lipshitz-Sarkar's Steenrod square operation on Khovanov homology coincides with Morán's cup-i operation, linking two different constructions of cohomology operations in knot theory.

Contribution

It establishes the equivalence of Lipshitz-Sarkar's and Morán's second Steenrod square operations on Khovanov homology, unifying two approaches to cohomology operations.

Findings

Lipshitz-Sarkar's Sq^2 matches Morán's mathfrak{sq}^2.

The first Steenrod operation not determined solely by homological data.

Bridges the gap between stable homotopy type and semi-simplicial approaches.

Abstract

Lipshitz-Sarkar defined a stable homotopy type refining Khovanov homology, producing cohomology operations $\text{Sq}^i$ on the Khovanov homology $Kh(L)$ of a link $L$. Later, Mor\'an proposed a sequence of cup-i products on the $\mathbb{F}_2$-coefficient cochain complex of any augmented semi-simplicial object in the Burnside category. Applied to the Khovanov functor, he obtained another sequence of operations $\mathfrak{sq}^n$ on $Kh(L)$, where $\mathfrak{sq}^0$, $\mathfrak{sq}^1$ agree with the usual Steenrod squares. We prove that Lipshitz-Sarkar's $\text{Sq}^2$, the first Steenrod operation that cannot be computed from merely homological data, agrees with Mor\'an's $\mathfrak{sq}^2$.