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gl(2) foams and the Khovanov homotopy type

Vyacheslav Krushkal, Paul Wedrich

Abstract

The Blanchet link homology theory is an oriented model of Khovanov homology, functorial over the integers with respect to link cobordisms. We formulate a stable homotopy refinement of the Blanchet theory, based on a comparison of the Blanchet and Khovanov chain complexes associated to link diagrams. The construction of the stable homotopy type relies on the signed Burnside category approach of Sarkar-Scaduto-Stoffregen.

gl(2) foams and the Khovanov homotopy type

Abstract

The Blanchet link homology theory is an oriented model of Khovanov homology, functorial over the integers with respect to link cobordisms. We formulate a stable homotopy refinement of the Blanchet theory, based on a comparison of the Blanchet and Khovanov chain complexes associated to link diagrams. The construction of the stable homotopy type relies on the signed Burnside category approach of Sarkar-Scaduto-Stoffregen.

Paper Structure

This paper contains 12 sections, 15 theorems, 25 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Blanchet's oriented model for Khovanov homology
  3. Blanchet's oriented model
  4. Bases for web spaces
  5. Construction of bases
  6. Action of foams on bases
  7. The stable homotopy type
  8. The signed Burnside category and the construction of ${\mathcal{X}}_{\rm or}(L)$
  9. Ladybug configurations and the completion of the construciton
  10. Reidemeister moves
  11. Maps induced by cobordisms
  12. Relation to the original construction of the Khovanov homotopy type

Key Result

Theorem 1.1

Let $L$ be an oriented link diagram. The stable homotopy type ${\mathcal{X}}_{\rm or}(L)$ is an invariant of the isotopy class of the link. Its cohomology is isomorphic to Blanchet's oriented model of Khovanov homology of $L$.

Figures (2)

  • Figure 2: Ladybug configuration
  • Figure 3: Reidemeister moves

Theorems & Definitions (40)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Theorem 2.5
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 30 more