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 [2] 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.

Figure 1

Index-0 (bottom row) and index-1 (top row) critical points, shown with their bond networks.

Figure 2

Bird's-eye view of the dendrogram for labeled disks. New components appear at radii 0.2071, 0.1964, and 0.1705 and components merge at radii 0.1942, 0.1871, and 0.1693.

Figure 3

Two saddle points with nearly the same radius.

Figure 4

For radius $0.1871<r<0.1942$, each connected component of the configuration space is homotopy equivalent to a circle. One such embedded circle and 20 index-0 critical points is shown here. Not shown are the 20 transitionary index-1 critical points between them.

Figure 5

Dendrogram for unlabeled disks. The numbers indicate the order in which critical points appear as the disks shrink—new components appear at 1, 2, and 5 and components merge at 3, 4, and 6. The radius where each vertex and edge appear is given in Fig. 8.

Figure 6

From left to right, critical submanifolds of dimensions 1, 2, 3, and 5.

Figure 7

Degenerate types.

Figure 8

Nondegenerate types, in rows by index. Index-0 critical points are in the bottom row, all the way up to index 4 in the top row.

Figure 9

Betti numbers for the first several changes in topology.

Figure 10

Histogram of the 19 types of nondegenerate critical points.

Figure 11

For four disks in a square, the topology changes eight times as the radius varies. Each row and column is unimodal and the maximum of each row is boxed.