Volume 17, issue 3 (2017)

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On bordered theories for Khovanov homology

Andrew Manion

Algebraic & Geometric Topology 17 (2017) 1557–1674
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 $\mathsc{ℬ}{\Gamma }_{n}$, 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.

Keywords
Khovanov homology, bordered Floer homology, invariants of tangles, linear-quadratic algebras
Primary: 57M27