Computational topology for configuration spaces of hard disks

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.

Min-type Morse theory for configuration spaces of hard spheres

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.