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On the functoriality of Khovanov homology

Dorra Hamza

TL;DR

Addresses the sign ambiguity in functoriality of Khovanov homology and provides a self contained exposition of Blanchet's oriented foam model yielding strict functoriality. It develops the foam category, the universal construction, and a Khovanov complex using foams, and relates it to a trivalent TQFT and the categorification of the Jones polynomial. It proves strict functoriality, recovers the original Khovanov homology, extends to Lee deformation, and to immersed cobordisms including singularities. The results solidify the functorial framework and enable robust topological invariants with applications to Rasmussen's $s$-invariant and 4-dimensional topology.

Abstract

We provide an overview of Blanchet's oriented model of Khovanov homology, which resolves the sign ambiguity in functoriality. We give a detailed and self-contained exposition of the construction. We establish strict functoriality for the oriented model and briefly outline its extension to cobordisms with singularities recently studied by Scott Carter, Benjamin Cooper, Mikhail Khovanov and Vyacheslav Krushkal.

On the functoriality of Khovanov homology

TL;DR

Addresses the sign ambiguity in functoriality of Khovanov homology and provides a self contained exposition of Blanchet's oriented foam model yielding strict functoriality. It develops the foam category, the universal construction, and a Khovanov complex using foams, and relates it to a trivalent TQFT and the categorification of the Jones polynomial. It proves strict functoriality, recovers the original Khovanov homology, extends to Lee deformation, and to immersed cobordisms including singularities. The results solidify the functorial framework and enable robust topological invariants with applications to Rasmussen's -invariant and 4-dimensional topology.

Abstract

We provide an overview of Blanchet's oriented model of Khovanov homology, which resolves the sign ambiguity in functoriality. We give a detailed and self-contained exposition of the construction. We establish strict functoriality for the oriented model and briefly outline its extension to cobordisms with singularities recently studied by Scott Carter, Benjamin Cooper, Mikhail Khovanov and Vyacheslav Krushkal.
Paper Structure (15 sections, 1 theorem)

This paper contains 15 sections, 1 theorem.

Table of Contents

  1. Introduction
  2. The foam category
  3. Evaluation of foams
  4. The universal construction
  5. Evaluation of graphs
  6. The Khovanov complex using foams
  7. Formal chain complexes in the category of foams
  8. The definition of the Khovanov complex: A picture
  9. Invariance
  10. The trivalent TQFT
  11. Categorification of the Jones polynomial
  12. Recovering the original Khovanov homology
  13. Lee Khovanov homology
  14. Functoriality
  15. Immersed cobordisms

Key Result

Theorem 6.1

Let $D$ be a diagram of a tangle $T$. Then:

Theorems & Definitions (2)

  • Theorem 6.1
  • proof