Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 081, 42 pages      arXiv:2210.06415      https://doi.org/10.3842/SIGMA.2023.081

Packing Densities of Delzant and Semitoric Polygons

Yu Du a, Gabriel Kosmacher a, Yichen Liu a, Jeff Massman a, Joseph Palmer ab, Timothy Thieme a, Jerry Wu a and Zheyu Zhang a
a)  Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA
b)  Department of Mathematics, University of Antwerp,Antwerp, Belgium

Received November 22, 2022, in final form October 20, 2023; Published online October 29, 2023

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.

Key words: equivariant packing; equivariant symplectic capacities; semitoric integrable systems; semitoric polygons; integrable systems.

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