Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Bar-Natan homotopy type

Taketo Sano

TL;DR

This work constructs a spatial refinement of Bar-Natan homology by producing a CW-spectrum $\mathcal{X}_{BN}(D)$ whose reduced cochain complex recovers the Bar-Natan complex, with the stable homotopy type $\mathcal{X}_{BN}(D)$ shown to be a link invariant. The principal construction passes through an XY-basis refinement $\mathcal{X}_{XY}(D)$, whose canonical cells capture Bar-Natan’s canonical generators, and then uses handle slides to realize a $1X$-based Bar-Natan flow category $\mathcal{C}_{BN}(D)$ whose spectrum $\mathcal{X}_{BN}(D)$ decomposes as a wedge of canonical cells. A central theme is lifting the Bar-Natan quantum filtration to the space level, potentially yielding a cohomotopical refinement of the $s$-invariant, though this requires controlling higher-dimensional moduli via Whitney-type moves. The paper also develops cobordism maps, duality, and canonical cohomotopy classes at the space level, and outlines future prospects including higher moduli spaces and Steenrod operations, which could yield stronger slice-genus bounds or new concordance invariants. Overall, it adapts the Khovanov homotopy-type framework to Bar-Natan theory, providing a robust space-level perspective and a roadmap toward a cohomotopical $s$-invariant.

Abstract

A spatial refinement of Bar-Natan homology is given, that is, for any link diagram $D$ we construct a CW-spectrum $\mathcal{X}_{\mathit{BN}}(D)$ whose reduced cellular cochain complex gives the Bar-Natan complex of $D$. The stable homotopy type of $\mathcal{X}_{\mathit{BN}}(D)$ is a link invariant and is described as the wedge sum of the canonical cells. We conjecture that the quantum filtration of Bar-Natan homology also lifts to the spatial level, and that it leads us to a cohomotopical refinement of the $s$-invariant.

A Bar-Natan homotopy type

TL;DR

This work constructs a spatial refinement of Bar-Natan homology by producing a CW-spectrum whose reduced cochain complex recovers the Bar-Natan complex, with the stable homotopy type shown to be a link invariant. The principal construction passes through an XY-basis refinement , whose canonical cells capture Bar-Natan’s canonical generators, and then uses handle slides to realize a -based Bar-Natan flow category whose spectrum decomposes as a wedge of canonical cells. A central theme is lifting the Bar-Natan quantum filtration to the space level, potentially yielding a cohomotopical refinement of the -invariant, though this requires controlling higher-dimensional moduli via Whitney-type moves. The paper also develops cobordism maps, duality, and canonical cohomotopy classes at the space level, and outlines future prospects including higher moduli spaces and Steenrod operations, which could yield stronger slice-genus bounds or new concordance invariants. Overall, it adapts the Khovanov homotopy-type framework to Bar-Natan theory, providing a robust space-level perspective and a roadmap toward a cohomotopical -invariant.

Abstract

A spatial refinement of Bar-Natan homology is given, that is, for any link diagram we construct a CW-spectrum whose reduced cellular cochain complex gives the Bar-Natan complex of . The stable homotopy type of is a link invariant and is described as the wedge sum of the canonical cells. We conjecture that the quantum filtration of Bar-Natan homology also lifts to the spatial level, and that it leads us to a cohomotopical refinement of the -invariant.

Paper Structure

This paper contains 22 sections, 50 theorems, 158 equations, 12 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Khovanov homology theory
  4. Resolution configurations
  5. Flow categories
  6. Flow category moves
  7. Basic construction
  8. $XY$-based Bar-Natan flow category
  9. Framing
  10. Determining the stable homotopy type
  11. Change of basis
  12. $1X$-based Bar-Natan flow category
  13. Quantum grading
  14. Cobordism maps
  15. Reidemsiter moves
  16. ...and 7 more sections

Key Result

Theorem 1

For any link diagram $D$, there is a CW-spectrum $\mathcal{X}_\mathit{BN}(D)$ whose reduced cellular cochain complex gives the Bar-Natan complex $C_{\mathit{BN}}(D)$. The cells of $\mathcal{X}_\mathit{BN}(D)$ correspond one-to-one to the standard generators of $C_{\mathit{BN}}(D)$.

Figures (12)

  • Figure 1: Strategy of construction
  • Figure 2: Coloring the Seifert circles by $\mathbf{a}$, $\mathbf{b}$.
  • Figure 3: A resolution configuration associated to a link diagram
  • Figure 4: The basic relation for $\mathcal{X}_{XY}$
  • Figure 5: An example of $D$ and its associated poset $P_{XY}(D)$
  • ...and 7 more figures

Theorems & Definitions (146)

  • Theorem 1
  • Theorem 2
  • Conjecture 1
  • Conjecture 2
  • Proposition 1.1: \ref{['prop:eliminate-0dim-moduli-spaces']}
  • Proposition 1.2: \ref{['def:s-bar', 'cor:bar-s-bound']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • ...and 136 more