On bordered theories for Khovanov homology

Abstract

We describe how to formulate Khovanov’s functor-valued invariant of tangles in the language of bordered Heegaard Floer homology. We then give an alternate construction of Lawrence Roberts’ type $D$ and type $A$ structures in Khovanov homology, and his algebra $\mathcal{ℬ}{\Gamma }_n$, in terms of Khovanov’s theory of modules over the ring $H^n$. We reprove invariance and pairing properties of Roberts’ bordered modules in this language. Along the way, we obtain an explicit generators-and-relations description of $H^n$ which may be of independent interest.