{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/configuration-spaces-of-disks-in-an-infinite","title":"Configuration spaces of disks in an infinite strip","arxiv_id":"1908.04241","date":"2019-08-12","proceeding":null,"authors":["Hannah Alpert","Matthew Kahle","Robert MacPherson"],"abstract":"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 $H_j [C(n,w)]$ behaves in qualitatively different ways. We show that if $w \\ge j+2$, then the homology $H_j[C(n, w)]$ is isomorphic to the homology of the configuration space of points in the plane, $H_j[C(n, \\mathbb{R}^2)]$. The Betti numbers of $C(n, \\mathbb{R}^2) $ were computed by Arnold, and so as a corollary of the isomorphism, $\\beta_j[C(n,w)]$ is a polynomial in $n$ of degree $2j$. On the other hand, we show that if $2 \\le w \\le j+1$, then $\\beta_j [ C(n,w) ]$ grows exponentially 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 homological \"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.","url_abs":"https://arxiv.org/abs/1908.04241v2","url_pdf":"https://arxiv.org/pdf/1908.04241v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"configuration-spaces-of-disks-in-an-infinite","repo_url":"https://github.com/burakericok/hard_disk_crits","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"configuration-spaces-of-disks-in-an-infinite","repo_url":"https://github.com/harharkh/HDCP","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}