Plane Floer homology and the odd Khovanov homology of 2-knots

Abstract

We prove a conjecture of Migdail and Wehrli regarding the odd Khovanov cobordism maps associated to knotted spheres. Our key tool is Daemi's plane Floer homology, which we use in place of a Lee deformation. Continuing the analogy with Lee homology, we see this work as a potential first step toward a genuinely functorial model for odd Khovanov homology.

Figures (24)

• Figure 1: Decorations indicating the choice of homology orientation on the split cobordism.
• Figure 2:
• Figure 3: Adding a crossing for the negative Reidemeister I move
• Figure 4: Removing a crossing for the positive Reidemeister I move
• Figure 5: Local picture of the plane knot complex for adding two crossings.

Theorems & Definitions (33)

• Theorem 1.1: migdail2024functoriality, Corollary 1
• Theorem 1.2
• Theorem 1.3: See daemi2015abelian, Theorem 4
• Proposition 1.4
• Proposition 1.5
• Remark
• Proposition 2.1: daemi2015abelian, Lemma 1.6
• Remark
• Proposition 2.2: daemi2015abelian, Proposition 1.22
• proof : Proof of Proposition \ref{['prop:step2']}