Computational topology for configuration spaces of hard disks

TL;DR

This paper investigates the topology of configuration spaces of hard disks through experimental methods, confirming theoretical predictions and proposing conjectures about their behavior as the number of disks increases.

Contribution

It provides experimental evidence for topological changes in disk configurations and supports a theorem relating critical points to balanced stresses, suggesting asymptotic behavior.

Findings

Topology changes observed with few disks

Critical points linked to balanced stresses

Conjectures on asymptotic topology

Abstract

We explore the topology of configuration spaces of hard disks experimentally, and show that several changes in the topology can already be observed with a small number of particles. The results illustrate a theorem of Baryshnikov, Bubenik, and Kahle that critical points correspond to configurations of disks with balanced mechanical stresses, and suggest conjectures about the asymptotic topology as the number of disks tends to infinity.