Configuration spaces of disks in an infinite strip

Abstract

We study the topology of the configuration spaces $\mathcal {C}(n,w)$ of n hard disks of unit diameter in an infinite strip of width w. We describe ranges of parameter or “regimes”, where homology $H_j [\mathcal {C}(n,w)]$ behaves in qualitatively different ways. We show that if $w \ge j+2$, then the homology $H_j[\mathcal {C}(n, w)]$ is isomorphic to the homology of the configuration space of points in the plane, $H_j[\mathcal {C}(n, \mathbb {R}^2)]$. The Betti numbers of $\mathcal {C}(n, \mathbb {R}^2)$ were computed by Arnold (The cohomology ring of the colored braid group. Springer Berlin, pp 183–186, 2014), and so as a corollary of the isomorphism, $\beta _j[\mathcal {C}(n,w)]$ is a polynomial in n of degree 2j. On the other hand, we show that if $2 \le w \le j+1$, then $\beta _j [ \mathcal {C}(n,w) ]$ grows exponentially with n. Most of our work is in carefully estimating $\beta _j [ \mathcal {C}(n,w) ]$ in this regime. We also illustrate, for every n, the homological “phase portrait” in the (w, j)-plane—the parameter values where homology $H_j [ \mathcal {C}(n,w)]$ is trivial, nontrivial, and isomorphic with $H_j [ \mathcal {C}(n, \mathbb {R}^2)]$. Motivated by the notion of phase transitions for hard-spheres systems, we discuss these as the “homological solid, liquid, and gas” regimes.

Appendix by Ulrich Bauer and Kyle Parsons

We computed the Betti numbers $\beta _j \left[ \mathrm{cell}(n,w) \right]$ for $n \le 8$, for homology with $\mathbb {Z}/2$ coefficients, using the software PHAT (Bauer et al. 2017b). The results of the computations appear in Table 1. For a point of reference, we note that $\mathrm{cell}(8)$ has over 5 million cells.

Table 1 Betti numbers of $\mathcal {C}(n,w)$ for small n and w