Min-type Morse theory for configuration spaces of hard spheres

TL;DR

This paper develops a Morse-theoretic framework to analyze the topology of configuration spaces of hard spheres, identifying critical points as balanced configurations and determining thresholds for topological equivalences.

Contribution

It introduces a novel Morse-theoretic approach to configuration spaces of hard spheres and characterizes critical points as balanced configurations.

Findings

Identifies the threshold radius for homotopy equivalence to point configurations.

Develops a general Morse-theoretic framework for hard sphere configurations.

Shows that balanced configurations serve as critical points in the topology analysis.

Abstract

We study configuration spaces of hard spheres in a bounded region. We develop a general Morse-theoretic framework, and show that mechanically balanced configurations play the role of critical points. As an application, we find the precise threshold radius for a configuration space to be homotopy equivalent to the configuration space of points.