[Paper Review] Odd Khovanov homology of principally unimodular bipartite graph-links

Igor Nikonov|arXiv (Cornell University)|Jun 1, 2010

Geometric and Algebraic Topology4 references3 citations

TL;DR

This paper introduces an integral version of odd Khovanov homology over ℤ for principally unimodular bipartite graph-links, overcoming sign ambiguities in intersection matrices by leveraging bipartite structure and principal unimodularity. The key contribution is proving that this homology theory is invariant under Reidemeister moves, establishing it as a topological invariant for this class of graph-links.

ABSTRACT

We define integral odd Khovanov homology of principally unimodular bipartite graph-links.

Motivation & Objective

• To define an integral (over ℤ) version of odd Khovanov homology for graph-links, overcoming sign ambiguities in intersection matrices.
• To restrict the definition to bipartite graph-links to preserve sign information of chord diagrams under mutations.
• To impose principal unimodularity to maintain orientation invariance under Reidemeister moves.
• To establish that the resulting homology is invariant under the four Reidemeister moves for this class of graph-links.
• To investigate whether this class of graph-links captures classical link invariants and whether non-realizable examples exist.

Proposed method

• Define oriented, labeled bipartite graphs with vertex signs and adjacency matrices over ℤ, using directed edges to encode orientation.
• Represent states as subgraphs and use block matrix structure (0, B; -B^T, 0) to reflect bipartite structure.
• Introduce four Reidemeister moves (R, Ω₁, Ω₂, Ω₃, Ω₄) on labeled oriented bipartite graphs to model isotopy.
• Construct chain complexes using exterior algebras over state spaces, with edge assignments to track sign data.
• Define chain maps between complexes under Reidemeister moves, showing compatibility via isomorphisms of edge assignments.
• Prove invariance by showing that induced maps on homology are isomorphisms, using anticommutation of 2-faces and cohomological triviality of collapsed complexes.

Experimental results

Research questions

• RQ1 Can odd Khovanov homology be defined over the integers for graph-links, preserving sign information from chord diagrams?
• RQ2 Why is the bipartite structure necessary to retain sign data under mutations?
• RQ3 How does principal unimodularity ensure orientation consistency under Reidemeister moves?
• RQ4 Is the resulting odd Khovanov homology invariant under all four Reidemeister moves for this class of graph-links?
• RQ5 Do bipartite principally unimodular graph-links provide a natural integral extension of classical link invariants?

Key findings

• Odd Khovanov homology over ℤ is well-defined for principally unimodular bipartite graph-links, resolving sign ambiguities via structural constraints.
• The homology theory is invariant under all four Reidemeister moves, as shown by constructing isomorphisms of chain complexes.
• The edge assignment system ensures consistency of signs across 2-faces, with anticommutation relations preserving chain complex structure.
• The complex remains isomorphic after Ω₄ moves due to a carefully defined map Φ that preserves face types and edge assignments.
• The quotient complex after collapsing faces has trivial H², enabling extension of edge assignments and proving invariance.
• The construction confirms that odd Khovanov homology is a topological invariant for this class of graph-links.