2019/08/12 by Alpert, Hannah, Kahle, Matthew, MacPherson, Robert · 2 citations
#55R80 #82B26 #Algebraic Topology (math.AT) #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1908.04241
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.