Strand algebras and contact categories

TL;DR

This paper establishes an isomorphism linking the homology of strand algebras in bordered Floer homology with contact category algebras, bridging Floer theory and contact topology through combinatorial and geometric correspondences.

Contribution

It introduces a novel isomorphism connecting strand algebra homology and contact category algebra, clarifying their relationship via arc diagrams and contact structures.

Findings

Correspondence between arc diagrams and quadrangulated surfaces

Strand diagrams correspond to contact structures

Stacking strand diagrams models stacking contact structures

Abstract

We demonstrate an isomorphism between the homology of the strand algebra of bordered Floer homology, and the category algebra of the contact category introduced by Honda. This isomorphism provides a direct correspondence between various notions of Floer homology and arc diagrams, on the one hand, and contact geometry and topology on the other. In particular, arc diagrams correspond to quadrangulated surfaces, idempotents correspond to certain basic dividing sets, strand diagrams correspond to contact structures, and multiplication of strand diagrams corresponds to stacking of contact structures. The contact structures considered are cubulated, and the cubes are shown to behave equivalently to local fragments of strand diagrams.