A Steenrod square on Khovanov homology (Q2921099)

In [J. Am. Math. Soc. 27, No. 4, 983--1042 (2014; Zbl 1345.57014)], the authors defined a CW complex, called Khovanov space, for Khovanov homology. The suspension spectrum of the space, called Khovanov spectrum and denoted by \(\mathcal{X}_{\mathrm{Kh}}(L)\), has the properties that its reduced cohomology coincides with Khovanov homology and its stable homotopy type is a link invariant. The current paper considers the stable cohomology operations on Khovanov homology, induced by \(\mathcal{X}_{\mathrm{Kh}}(L)\).NEWLINENEWLINEIn particular, the paper is devoted to giving a pure combinatorial description of the Steenrod square maps \(\mathrm{Sq}^{1}: \mathrm{Kh}^{i,j}_{\mathbb{F}_{2}}(L) \to \mathrm{Kh}^{i+1,j}_{\mathbb{F}_{2}}(L)\) and \(\mathrm{Sq}^{2}: \mathrm{Kh}^{i,j}_{\mathbb{F}_{2}}(L) \to \mathrm{Kh}^{i+2,j}_{\mathbb{F}_{2}}(L)\). Section 2 provides explicit definitions for the maps, and Section 3 proves that \(\mathrm{Sq}^{2}\) defined in Section 2 is indeed the Steenrod square.NEWLINENEWLINEIn Section 4, the authors prove that under some conditions on \mathrm{Kh}ovanov homology, the homotopy type of \(\mathcal{X}_{\mathrm{Kh}}(L)\) is determined by the maps \(\mathrm{Sq}^{1}\) and \(\mathrm{Sq}^{2}\). In the end, they show that all links up to 11 crossings satisfy the conditions.