Functoriality for the $\mathfrak{su}_3$ Khovanov homology

Abstract

We prove that the categorified ${\mathfrak{s}\mathfrak{u}}_3$ quantum link invariant is functorial with respect to tangle cobordisms. This is in contrast to the categorified ${\mathfrak{s}\mathfrak{u}}_2$ theory, which was not functorial as originally defined.

We use methods of Morrison and Nieh and 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, up to homotopy, under Carter and Saito’s movie moves.