Packing Densities of Delzant and Semitoric Polygons

TL;DR

This paper develops methods to compute packing densities and capacities of 4D toric and semitoric integrable systems using their polygons, providing new results and solutions to the perfect packing problem.

Contribution

It introduces techniques to calculate equivariant packing densities directly from polygons, expanding previous results and solving the semitoric perfect packing problem.

Findings

Computed densities for key examples

Solved the equivariant semitoric perfect packing problem

Provided an accessible introduction to Delzant and semitoric polygons

Abstract

Exploiting the relationship between 4-dimensional toric and semitoric integrable systems with Delzant and semitoric polygons, respectively, we develop techniques to compute certain equivariant packing densities and equivariant capacities of these systems by working exclusively with the polygons. This expands on results of Pelayo and Pelayo-Schmidt. We compute the densities of several important examples and we also use our techniques to solve the equivariant semitoric perfect packing problem, i.e., we list all semitoric polygons for which the associated semitoric system admits an equivariant packing which fills all but a set of measure zero of the manifold. This paper also serves as a concise and accessible introduction to Delzant and semitoric polygons in dimension four.