Polytopes, dualities, and Floer homology

This article is an exposition of a body of existing results, together with an announcement of recent results. We discuss a theory of polytopes associated to bipartite graphs and trinities, developed by Kálmán, Postnikov and others. This theory exhibits a variety of interesting duality and triality relations, and extends into knot theory, 3-manifold topology and Floer homology. In recent joint work with Kálmán, we extend this story into contact topology and contact invariants in sutured Floer homology.

Strand algebras and contact categories

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.