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Configuration spaces of disks in an infinite strip

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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 (wj)-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.

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Correspondence to Matthew Kahle.

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Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

M.K. thanks IAS for hosting him as a member in 2010–11, and for several visits since then. He gratefully acknowledges the support of a Sloan Research Fellowship, NSF #DMS-1352386, and NSF #CCF-1839358. H.A. is supported by the National Science Foundation under Award No. DMS 1802914.

All three authors thank ICERM for hosting them during the special thematic semester “Topology in Motion” in Autumn 2016.

Appendix by Ulrich Bauer and Kyle Parsons

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

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Alpert, H., Kahle, M. & MacPherson, R. Configuration spaces of disks in an infinite strip. J Appl. and Comput. Topology 5, 357–390 (2021).

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