Configuration spaces of disks in an infinite strip

Hannah Alpert, Matthew Kahle, Robert MacPherson

Published 2019-08-12Version 1

We study the topology of the configuration space $C(n,w)$ of $n$ hard disks of unit diameter in an infinite strip of width $w$. We describe ranges of parameter or "regimes", where homology $H_j [C(n,w)]$ behaves in qualitatively different ways. We show that if $w \ge j+2$, then the inclusion $i$ into the configuration space of $n$ points in the plane $C(n,\mathbb{R}^2)$ induces an isomorphism on homology $i_{*} : H_j [C(n,w) ] \to H_j [ C(n,\mathbb{R}^2)]$. The Betti numbers of $C(n, \mathbb{R}^2) $ were computed by Arnold, and so as a corollary of the isomorphism, if $w$ and $j$ are fixed then $\beta_j[C(n,w)]$ is a polynomial of degree $2j$ in $n$. On the other hand, we show that $w$ and $j$ are fixed and $2 \le w \le j+1$, then $\beta_j [ C(n,w) ]$ grows exponentially fast with $n$. Most of our work is in carefully estimating $\beta_j [ C(n,w) ]$ in this regime. We also illustrate for every $n$ the "phase portrait" in the $(w,j)$-plane--- the parameter values where homology $H_j [ C(n,w)]$ is trivial, nontrivial, and isomorphic with $H_j [ C(n, \mathbb{R}^2)]$. Motivated by the notion of phase transitions for hard-spheres systems, we discuss these as the "homological solid, liquid, and gas" regimes.