A refinement of Rasmussen’s S-invariant

Abstract

In a previous work, we constructed a spectrum-level refinement of Khovanov homology. This refinement induces stable cohomology operations on Khovanov homology. In this paper we show that these cohomology operations commute with cobordism maps on Khovanov homology. As a consequence we obtain a refinement of Rasmussen’s slice genus bound $s$ for each stable cohomology operation. We show that in the case of the Steenrod square ${Sq}^2$ our refinement is strictly stronger than $s$.