Fixing the functoriality of Khovanov homology: a simple approach

TL;DR

The paper addresses sign indeterminacy in the functoriality of Khovanov-type homology under link cobordisms by staying within Bar-Natan's formal complexes and introducing a sign-adjustment scheme for Reidemeister and Morse moves. Using the universal theory $H_{h,t}$ and its specialization, it derives explicit behavior of canonical cycles $\alpha(D)$ and $\beta(D)$ under cobordism maps, producing invariance under movie moves and a new closed-surface invariant. The approach yields a concrete, self-contained route to strict functoriality up to chain homotopy without foam or seamed-cobordism extensions, applicable across Khovanov-type theories. It also recovers a Khovanov–Jacobsson-type invariant for closed surfaces and clarifies the relationship to existing invariants like Rasmussen and Tanaka.

Abstract

Khovanov homology is functorial up to sign with respect to link cobordisms. The sign indeterminacy has been fixed by several authors, by extending the original theory both conceptually and algebraically. In this paper we propose an alternative approach: we stay in the classical setup and fix the functoriality by simply adjusting the signs of the morphisms associated to the Reidemeister moves and the Morse moves.

Figures (3)

• Figure 1: Coloring the Seifert circles by $\mathbf{a}$, $\mathbf{b}$.
• Figure 2: R-III move
• Figure :

Theorems & Definitions (27)

• Theorem 1
• Corollary 1.1
• Theorem 2
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Remark 2.6
• Definition 2.9: $\alpha$-, $\beta$-cycles