On bordered theories for Khovanov homology

TL;DR

This paper reformulates Khovanov homology invariants within bordered Heegaard Floer homology, providing new constructions and explicit descriptions that enhance understanding and computational approaches.

Contribution

It introduces an alternative construction of Roberts' bordered structures in Khovanov homology using modules over the ring H^n, and offers an explicit generators-and-relations description of H^n.

Findings

Reproves invariance and pairing properties of Roberts' bordered modules.

Provides an explicit algebraic description of H^n.

Connects Khovanov homology with bordered Heegaard Floer homology frameworks.

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.