Polytopes, dualities, and Floer homology

TL;DR

This paper reviews and extends a theory connecting polytopes, dualities, and Floer homology, highlighting recent advances in contact topology and invariants in sutured Floer homology.

Contribution

It synthesizes existing results on polytopes associated with bipartite graphs and introduces new connections to contact topology and sutured Floer homology.

Findings

Development of polytope theories linked to bipartite graphs and trinities

Discovery of duality and triality relations in these theories

Extension of the framework into contact topology and invariants in sutured Floer homology

Abstract

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 K\'alm\'an, 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 K\'alm\'an, we extend this story into contact topology and contact invariants in sutured Floer homology.