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Spectra in Khovanov and knot Floer theories

Marco Marengon, Sucharit Sarkar, Andras Stipsicz

TL;DR

This note surveys the construction of stable homotopy types for two categorified knot invariants: the Khovanov stable homotopy type (Lipshitz–Sarkar) and a knot Floer stable homotopy type (Manolescu–Sarkar). It develops spacification via framed flow categories and the Cohen–Jones–Segal construction, starting from chain complexes and moving to spectra whose cellular chains recover the original homology. For Khovanov, the cube of resolutions yields moduli spaces that are permutohedra, with ladybug matchings resolving 1-dimensional ambiguities to produce a coherent stable homotopy type whose homology is Khovanov homology. For knot Floer, grid diagrams give grid homology and a more delicate moduli-space construction that must accommodate bubbles through obstruction theory and stratified gluing, culminating in a spectrum with homology equal to grid homology and opening the way to refinements via Steenrod operations and $K$-theory; several invariance and technical issues, particularly on the knot Floer side, remain active areas of study.

Abstract

These notes provide an introduction to the stable homotopy types in Khovanov theory (due to Lipshitz-Sarkar) and in knot Floer theory (due to Manolescu-Sarkar). They were written following a lecture series given by Sucharit Sarkar at the Renyi Institute during a special semester on "Singularities and low-dimensional topology", organised by the Erdos Center.

Spectra in Khovanov and knot Floer theories

TL;DR

This note surveys the construction of stable homotopy types for two categorified knot invariants: the Khovanov stable homotopy type (Lipshitz–Sarkar) and a knot Floer stable homotopy type (Manolescu–Sarkar). It develops spacification via framed flow categories and the Cohen–Jones–Segal construction, starting from chain complexes and moving to spectra whose cellular chains recover the original homology. For Khovanov, the cube of resolutions yields moduli spaces that are permutohedra, with ladybug matchings resolving 1-dimensional ambiguities to produce a coherent stable homotopy type whose homology is Khovanov homology. For knot Floer, grid diagrams give grid homology and a more delicate moduli-space construction that must accommodate bubbles through obstruction theory and stratified gluing, culminating in a spectrum with homology equal to grid homology and opening the way to refinements via Steenrod operations and -theory; several invariance and technical issues, particularly on the knot Floer side, remain active areas of study.

Abstract

These notes provide an introduction to the stable homotopy types in Khovanov theory (due to Lipshitz-Sarkar) and in knot Floer theory (due to Manolescu-Sarkar). They were written following a lecture series given by Sucharit Sarkar at the Renyi Institute during a special semester on "Singularities and low-dimensional topology", organised by the Erdos Center.
Paper Structure (17 sections, 2 theorems, 35 equations, 37 figures)

This paper contains 17 sections, 2 theorems, 35 equations, 37 figures.

Table of Contents

  1. Introduction
  2. Framed flow categories
  3. A recap of Morse theory
  4. From Morse functions to framed flow categories
  5. The hypercube flow category
  6. The Cohen-Jones-Segal construction
  7. A Khovanov stable homotopy type
  8. The Khovanov complex
  9. The Lipshitz-Sarkar framed flow category
  10. A knot Floer stable homotopy type
  11. Grid diagrams
  12. Grid homology
  13. Maslov index 1 domains
  14. Maslov index 2 domains
  15. Maslov index 3 domains
  16. ...and 2 more sections

Key Result

Lemma 4.4

The pair $(CD _*, \partial )$ is a chain complex, that is, $\partial ^2=0$.

Figures (37)

  • Figure 1: Schematic of a proper Morse function.
  • Figure 2: An illustration of the equality $T\mathcal{M}(x,y) \oplus \mathbb{R} \oplus T\mathbb{D}_y = T\mathbb{D}_x$.
  • Figure 3: Illustration of the flowlines from $11$ to $00$ and of the resulting moduli space.
  • Figure 4: The moduli space $\mathcal{M}(111,000)$.
  • Figure 5: A choice of orientation of the 0-dimensional moduli spaces.
  • ...and 32 more figures

Theorems & Definitions (14)

  • Remark 2.1
  • Remark 2.2
  • Definition 2.3
  • Example 2.4
  • Example 2.5
  • Definition 2.7
  • Definition 2.8
  • Remark 3.1
  • Remark 4.1
  • Remark 4.2
  • ...and 4 more