A Steenrod square on Khovanov homology and a cup-i product

Abstract

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 $\mathfrak{sq}^2$.

Figures (7)

• Figure 1: Left: the elements $s,t$ in the span $\partial_{ab} = \partial_{ab}(z)$ The thick arrows denote the spans $\partial^{n+2}_{a},\partial^{n+2}_{b},\partial^{n+1}_{b-1},\partial^{n+1}_{a}$, . Right: We write both $s$ and $t$ as compositions in $\partial_{b-1}\circ\partial_a$ (composing along the left side of the diagram) or in $\partial_{a}\circ\partial_b$ (composing along the right side of the diagram).
• Figure 2: We can write a span element $s:z\to x$ either as a composition in $\partial_b$.
• Figure 3: Left: A degenerate chord presentation $\overline{\mathcal{C}}(z,\alpha)$, for $\alpha$ a cocycle. There are two chords between $b$ and $e$, so the chord is labeled with a "$2$" to show multiplicity. Right: A chord presentation $\mathcal{C}(z,\alpha)$. Note the even number of chords coming out of each index.
• Figure 4: The types of chord pairs in $\overline{\mathcal{C}}(z,\alpha)$ counted in the interior summand of Equation (\ref{['ab+cd rewritten']}). Left: a chord pair of type \ref{['type 1']}. Middle: a chord pair of type \ref{['type 2r']}. Right: a chord pair of type \ref{['type 2l']}.
• Figure 5: We draw two cycles of chords (one solid, one dotted) corresponding to two graph cycles $K,K'\subset \Gamma(z,\alpha)$. Note how these two cycles have even intersection number, as they always should. We compute $\#(\mathcal{C}(K)\cap\mathcal{C}(K'))$ by adding the internal crossings in each cycle.

Theorems & Definitions (54)

• Theorem 1.1
• Corollary 1.2
• Definition 2.1
• Definition 2.2
• Definition 2.4
• Theorem 2.5: MR3252965,rajapakse2025
• Definition 2.7
• Remark 2.7.1
• Definition 3.3
• Definition 3.4