# 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.

## Key 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.