Functoriality for the su(3) Khovanov homology

TL;DR

This paper proves that the categorification of the su(3) quantum knot invariant is functorial with respect to tangle cobordisms, using explicit chain maps and invariance under movie moves, contrasting with the non-functorial su(2) case.

Contribution

It establishes functoriality for the su(3) Khovanov homology, providing explicit chain maps and invariance proofs, which was not known before.

Findings

Proves functoriality of su(3) Khovanov homology.

Constructs explicit chain maps for Reidemeister moves.

Shows invariance under Carter-Saito movie moves.

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.