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.