Papers › Configuration spaces of disks in an infinite strip
Configuration spaces of disks in an infinite strip
Hannah Alpert, Matthew Kahle, Robert MacPherson
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We study the topology of the configuration spaces 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 Hⱼ [C(n,w)] behaves in qualitatively different ways. We show that if w ≥j+2, then the homology Hⱼ[C(n, w)] is isomorphic to the homology of the configuration space of points in the plane, Hⱼ[C(n, ℝ²)]. The Betti numbers of C(n, ℝ²) were computed by Arnold, and so as a corollary of the isomorphism, βⱼ[C(n,w)] is a polynomial in n of degree 2j. On the other hand, we show that if 2 ≤w ≤j+1, then βⱼ [ C(n,w) ] grows exponentially with n. Most of our work is in carefully estimating βⱼ [ 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 Hⱼ [C(n,w)] is trivial, nontrivial, and isomorphic with Hⱼ [C(n, ℝ²)]. 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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