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

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 $\mathfrak{sq}^2$.