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Two second Steenrod squares for odd Khovanov homology

Dirk Schuetz

TL;DR

The paper establishes a framework to define and compute a second Steenrod square on odd Khovanov homology using framed 1-flow categories and signed Burnside categories, yielding link-invariant operations $\\mathrm{Sq}^2_\\varepsilon$ and refinements of Rasmussen-type concordance invariants $s_±^{\\mathrm{Sq}^2_\\varepsilon}$. It develops a cube-category based construction, proves naturality and independence from many choices, and relates the new squares to existing stable-homotopy constructions. The authors provide explicit calculations for knots, showing nontrivial Sq$^2$ in several cases and examining behavior under split unions and mirror duality, revealing nuanced relationships to Spanier–Whitehead duality. This work broadens the toolbox for distinguishing knots and concordance classes via cohomology operations in odd Khovanov theory and offers computational methods via KnotJob to explore these invariants. The results suggest rich interactions between odd Khovanov homology, Steenrod operations, and stable-homotopy-type refinements with potential applications to knot concordance and 4-manifold topology.

Abstract

Recently, Sarkar-Scaduto-Stoffregen constructed a stable homotopy type for odd Khovanov homology, hence obtaining an action of the Steenrod algebra on Khovanov homology with $\mathbb{Z}/2\mathbb{Z}$ coefficients. Motivated by their construction we propose a way to compute the second Steenrod square. Our construction is not unique, but we can show it to be a link invariant which gives rise to a refinement of the Rasmussen $s$-invariant with $\mathbb{Z}/2\mathbb{Z}$ coefficients. We expect it to be related to the second Steenrod square arising from the Sarkar-Scaduto-Stoffregen construction.

Two second Steenrod squares for odd Khovanov homology

TL;DR

The paper establishes a framework to define and compute a second Steenrod square on odd Khovanov homology using framed 1-flow categories and signed Burnside categories, yielding link-invariant operations and refinements of Rasmussen-type concordance invariants . It develops a cube-category based construction, proves naturality and independence from many choices, and relates the new squares to existing stable-homotopy constructions. The authors provide explicit calculations for knots, showing nontrivial Sq in several cases and examining behavior under split unions and mirror duality, revealing nuanced relationships to Spanier–Whitehead duality. This work broadens the toolbox for distinguishing knots and concordance classes via cohomology operations in odd Khovanov theory and offers computational methods via KnotJob to explore these invariants. The results suggest rich interactions between odd Khovanov homology, Steenrod operations, and stable-homotopy-type refinements with potential applications to knot concordance and 4-manifold topology.

Abstract

Recently, Sarkar-Scaduto-Stoffregen constructed a stable homotopy type for odd Khovanov homology, hence obtaining an action of the Steenrod algebra on Khovanov homology with coefficients. Motivated by their construction we propose a way to compute the second Steenrod square. Our construction is not unique, but we can show it to be a link invariant which gives rise to a refinement of the Rasmussen -invariant with coefficients. We expect it to be related to the second Steenrod square arising from the Sarkar-Scaduto-Stoffregen construction.
Paper Structure (21 sections, 18 theorems, 129 equations, 2 figures, 1 table)

This paper contains 21 sections, 18 theorems, 129 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. $1$-Flow Categories
  3. The cube category
  4. Framed 1-flow categories
  5. The second Steenrod Square of a framed 1-flow category
  6. Short exact sequences of $1$-flow categories
  7. Signed covers of the cube $1$-flow category
  8. The signed Burnside category
  9. The definition of the signed Burnside category
  10. Functors from the cube category to the signed Burnside category
  11. Odd Khovanov Homology
  12. A quick recollection of odd Khovanov homology
  13. Second Steenrod squares for odd Khovanov homology
  14. Reduced odd Khovanov homology
  15. Concordance Invariants
  16. ...and 6 more sections

Key Result

Proposition 1.1

Let $F_\sigma\colon \underline{2}^n\to\mathcal{B}_\sigma$ be a strictly unitary $2$-functor and $\varepsilon\in \mathbb{Z}/2\mathbb{Z}$. Then there exist linear maps which are non-zero in general, and can depend on $\varepsilon$.

Figures (2)

  • Figure 1: A crossing with $1$-smoothing on the right.
  • Figure 2: Commutation chart: Thick lines represent components in $S_{c}$, thin lines the surgery arcs. If a surgery arc has no orientation, then both orientations lead to the same result.

Theorems & Definitions (55)

  • Proposition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Example 2.6
  • Definition 2.7
  • ...and 45 more