Singular TQFTs, foams and type D arc algebras

TL;DR

This paper provides a combinatorial description of singular cobordisms, called foams, which underpin functorial Khovanov homology, and uses this to realize the type D arc algebra topologically.

Contribution

It introduces a combinatorial model for singular cobordisms (foams) and applies it to realize the type D arc algebra topologically.

Findings

Describes the 2-category of singular cobordisms (foams)

Realizes the type D arc algebra via singular cobordism construction

Provides a topological interpretation of the type D arc algebra

Abstract

We combinatorially describe the $2$-category of singular cobordisms, called (rank one) foams, which governs the functorial version of Khovanov homology. As an application we topologically realize the type $\mathrm{D}$ arc algebra using this singular cobordism construction.