A–infinity algebras, strand algebras, and contact categories

In previous work we showed that the contact category algebra of a quadrangulated surface is isomorphic to the homology of a strand algebra from bordered sutured Floer theory. Being isomorphic to the homology of a differential graded algebra, this contact category algebra has an A–infinity structure, allowing us to combine contact structures not just by gluing, but also by higher-order operations. We investigate such A–infinity structures and higher-order operations on contact structures. We give explicit constructions of such A–infinity structures, and establish some of their properties, including conditions for the vanishing and nonvanishing of A–infinity operations. Along the way we develop several related notions, including a detailed consideration of tensor products of strand diagrams.