[Paper Review] Link homology and unoriented topological quantum field theory

Vladimir Turaev, Paul Turner|arXiv (Cornell University)|Jun 13, 2005

Geometric and Algebraic Topology10 references5 citations

TL;DR

This paper introduces new link homology theories for stable equivalence classes of link diagrams on oriented surfaces by extending Bar-Natan’s geometric formalism using unoriented 1+1-dimensional topological quantum field theories (TQFTs). The key contribution is a systematic construction of link homology invariants that are invariant under stable equivalence, generalizing Khovanov homology to surfaces via unoriented TQFT structures.

ABSTRACT

ABSTRACT. We investigate Khovanov homology of stable equivalence classes of link diagrams on oriented surfaces. We apply Bar-Natan’s geometric formalism to this setting and using unoriented 1+1-dimensional topological quantum field theories we define new link homology theories for stable equivalences classes. 1.

Motivation & Objective

• To extend Khovanov homology to stable equivalence classes of link diagrams on oriented surfaces.
• To develop a geometric framework for link invariants in the context of surface-embedded links.
• To apply unoriented 1+1-dimensional topological quantum field theories (TQFTs) to construct new homology theories.
• To generalize Bar-Natan’s formalism to unoriented TQFTs in order to define invariants under stable equivalence.
• To establish a link homology theory that is invariant under stabilization moves on surfaces.

Proposed method

• Adapting Bar-Natan’s geometric formalism to the setting of link diagrams on oriented surfaces.
• Using unoriented 1+1-dimensional TQFTs as the algebraic backbone for defining homology groups.
• Defining chain complexes from link diagrams via cobordism categories over surfaces.
• Constructing invariants that are preserved under stable equivalence via TQFT functors.
• Employing state-sum models based on TQFT evaluations of cobordisms between link diagrams.
• Ensuring invariance under Kirby moves and stabilization by leveraging the structure of unoriented TQFTs.

Experimental results

Research questions

• RQ1 How can Khovanov homology be generalized to stable equivalence classes of links on oriented surfaces?
• RQ2 What role do unoriented 1+1-dimensional TQFTs play in constructing link invariants on surfaces?
• RQ3 In what ways does Bar-Natan’s geometric formalism extend to the unoriented and surface-embedded setting?
• RQ4 How do stabilization moves affect the homology invariants, and can they be made invariant under such moves?
• RQ5 What algebraic structures (e.g., TQFTs) are necessary and sufficient to define stable link homology theories on surfaces?

Key findings

• The paper constructs new link homology theories that are invariant under stable equivalence of link diagrams on oriented surfaces.
• Unoriented 1+1-dimensional TQFTs provide the necessary algebraic structure to define these homology theories in the surface setting.
• The extension of Bar-Natan’s formalism to unoriented TQFTs enables a geometric and algebraic framework for surface-embedded links.
• The resulting homology theories are well-defined and stable under stabilization, generalizing classical Khovanov homology.
• The construction establishes a bridge between surface topology, unoriented TQFTs, and link homology invariants.
• The framework allows for the systematic assignment of homology groups to link diagrams via cobordism categories and TQFT functors.