Steenrod square for virtual links toward Khovanov-Lipshitz-Sarkar stable homotopy type for virtual links

We define a second Steenrod square for virtual links, which is stronger than Khovanov homology for virtual links, toward constructing Khovanov-Lipshitz-Sarkar stable homotopy type for virtual links. This induces the first meaningful nontrivial example of the second Steenrod square operator on the Khovanov homology for links in a 3-manifold other than the 3-sphere.