arXiv subjects
Publications and source records attributed to Advika Rajapakse.
For an arbitrary link $L \subset S^3$ , Sarkar-Scaduto-Stoffregen construct a family of spatial refinements of even and odd Khovanov homology. We give a computation of $\text{Sq}^2$ on these spaces, determining their stable homotopy types for all knots K up to 11 crossings. We also prove that the Steenrod squares $\text{Sq}_0^2$ , $\text{Sq}_1^2$ defined by Schütz do arise as Steenrod squares on these spaces.
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, Morán 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 Morán's $\mathfrak{sq}^2$.