Singular crossings and Ozsv\'ath-Szab\'o's Kauffman-states functor

Recently, Ozsv'ath and Szab'o introduced some algebraic constructions computing knot Floer homology in the spirit of bordered Floer homology, including a family of algebras B(n) and, for a generator of the braid group on n strands, a certain type of bimodule over B(n). We define analogous bimodules for singular crossings. Our bimodules are motivated by counting holomorphic disks in a bordered sutured version of a Heegaard diagram considered previously by Ozsv'ath, Stipsicz, and Szab'o.