Higher Steenrod squares for Khovanov homology

TL;DR

This paper develops explicit formulas for higher Steenrod squares acting on Khovanov homology, providing new tools for understanding link invariants through cohomology operations.

Contribution

It introduces stable cup-i products for semi-simplicial objects in the Burnside category, enabling explicit calculations of Steenrod squares on Khovanov homology.

Findings

Explicit formulas for Steenrod squares on Khovanov homology.

Application to any link's Khovanov homology.

Enhanced understanding of cohomology operations in link invariants.

Abstract

We describe stable cup-i products on the cochain complex with $F^2$ coefficients of any augmented semi-simplicial object in the Burnside category. An example of such an object is the Khovanov functor of Lawson, Lipshitz and Sarkar. Thus we obtain explicit formulas for cohomology operations on the Khovanov homology of any link.