Functoriality for the su(3) Khovanov homology

Abstract: We prove that Morrison and Nieh's categorification of the su(3) quantum knot invariant is functorial with respect to tangle cobordisms. This is in contrast to the categorified su(2) theory, which was not functorial as originally defined. We use methods of Bar-Natan to construct explicit chain maps for each variation of the third Reidemeister move. Then, to show functoriality, we modify arguments used by Clark, Morrison, and Walker to show that induced chain maps are invariant under Carter and Saito's movie moves.

• The paper verifies functoriality in the categorification of the su(3) quantum knot invariant through explicit chain maps and homotopy constructions.
• It employs planar algebra techniques and canopoleis to lift Kuperberg’s spider to a decorated web and foam framework, ensuring consistent evaluations.
• Resolving sign anomalies present in su(2) cases, the work paves the way for robust TQFT applications and extensions to higher-dimensional invariants.

Functoriality in the $su(3)$ Khovanov Homology

Introduction and Context

The paper "Functoriality for the $su(3)$ Khovanov homology" (0806.0601) provides a rigorous verification of functoriality in the categorification of the $su(3)$ quantum knot invariant developed by Morrison and Nieh. In the landscape of link homology theories, functoriality with respect to link cobordisms is an essential property that elevates the theory from a mere link invariant to a bona fide topological quantum field theory (TQFT)-like structure. Unlike the original $su(2)$ Khovanov homology, where functoriality was obstructed by sign anomalies, the $su(3)$ situation, as proved in this paper, exhibits true functoriality without such ambiguities.

Planar Algebras, Canopoleis, and Link Cobordism

The formalism of planar algebras, essential for handling the local-to-global compositions in tangle diagrams, is upgraded in this context to the categorical level: canopoleis, which can be viewed as colored planar algebras internal to categories, with formal boundary labels encoding compatibility conditions for composition. The underlying motivation is the operative action of Kuperberg’s $su(3)$ spider within the planar algebra context, and the subsequent categorification involves not just graphs (webs) but surfaces termed foams, serving as morphisms in the categorified setting.

The source canopolis, $Ortang$, consists of tangles and their cobordisms, while the target, denoted as ${3}$, consists of complexes of decorated $su(3)$ webs with morphisms corresponding to foams modulo certain local relations. The formal structure ensures that planar diagrammatic manipulations correspond to functorial compositions of chain complexes.

Categorification and the Construction of the Theory

The crux of the $su(3)$ categorification is to lift the $q$-deformed skein-theoretic relations (from Kuperberg’s spider) to a homological framework. Formal complexes assigned to link (or tangle) diagrams are generically built by resolving each crossing as per the categorified skein relation, leading to objects that are direct sums of webs with grading and differentials given by specific foams (zip, unzip, saddle, etc.)—as established by Morrison and Nieh.

A key technical element is the intricate system of local relations imposed on foams—closed foam relations, neck cutting, airlock, tube, three-rocket, and seam-swap relations, among others—which guarantee that the theory correctly extends the $su(3)$ spider and that all closed foams are evaluated consistently.

Explicit Construction of Reidemeister and Cobordism-Induced Maps

A significant technical contribution of the paper is the explicit construction of chain maps induced by all variations of the oriented Reidemeister moves, including the full classification of oriented Reidemeister three (R3) moves. The combinatorics and algebraic structure here are complicated by the orientation and directionality of strands, the trivalent vertex structure of webs, and the required compatibilities of grading and orientation in the definition of foams.

The implementation uses a modular reduction via strong deformation retracts and repeated application of the cone construction for chain complexes, notably leveraging Bar-Natan’s homological algebraic techniques. The computations explicitly track the behavior of chain maps and manage all possible crossing orderings, ensuring the required sign conventions are consistently enforced throughout.

Functoriality and Movie Moves

Functoriality with respect to link (or tangle) cobordism is equivalent to naturality of the construction under movie moves—local modifications of tangle movies as classified by Carter and Saito. The verification that the categorified invariant does not depend on the chosen movie presentation (sequence of Reidemeister and Morse moves) but only on the isotopy type of the cobordism is the central technical content. The analysis comprises:

• Explicit chain homotopy equivalence for each movie move variation, including all necessary sign and grading considerations.
• Homotopy isolation techniques, duality, and degree bounds to facilitate computations and reduce the cases where explicit calculation is required.
• Complete analysis of non-reversible (Morse) movie moves via direct computation, and closure arguments for higher movie move variations via the geometry of the tangle projection stratification.

A critical distinction from the $su(2)$ case is the absence of sign anomalies; in $su(3)$, self-duality is lost, precluding the sign ambiguity that plagued $su(2)$ functoriality. The argument is therefore more direct, but still requires intricate combinatorial handling of all move variations.

Numerical and Structural Highlights

• Theorem: The categorified $su(3)$ link invariant $Kh(su(3))$ defines a functor from the canopolis of oriented tangles and cobordisms to the canopolis of formal complexes of $su(3)$ webs—i.e., it is functorial up to homotopy (0806.0601).
• Strong claim: The functoriality holds for all oriented tangle cobordisms and all movie move variations, without the sign ambiguity present in $su(2)$ Khovanov homology.
• Explicit chain maps and homotopies are identified for all variants, providing a computational foothold for further generalization and computer-assisted calculations.

Implications and Future Directions

This result secures the $su(3)$ categorified invariant as a candidate for extended TQFT structure and opens the possibility of extracting four-dimensional invariants or mapping class group representations from the theory. The machinery demonstrated here suggests that the structural hurdles endemic to categorified $su(n)$ invariants for $n>2$ can be overcome, at least for $su(3)$.

The natural progression is towards an extension to knotted webs and seamed cobordisms in higher genus and four-dimensional settings, with the challenge of additional vertex moves and the need for a renormalization of the skein relations and potentially the introduction of framing. The framework of canopoleis, as clarified here, provides an appropriately flexible categorical environment for such generalizations.

Conclusion

The paper establishes that the Morrison–Nieh categorification of the $su(3)$ link invariant is genuinely functorial with respect to oriented tangle cobordisms, represented concretely by movie moves. The proof leverages homological algebraic techniques, explicit computation of chain maps, and coherence checks across all movie moves, affirming both theoretical soundness and computational applicability. As a consequence, the $su(3)$ Khovanov homology stands as a robust categorified quantum invariant, ripe for applications and further extension in higher representation theoretic or TQFT contexts.