{"paper":{"title":"Vietoris-Rips complexes of torus grids","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AT","authors_text":"Adenike Yeside Adetowubo, Hector Barriga-Acosta, Henry Adams, John Sterling, Ziqin Feng","submitted_at":"2025-02-10T23:53:44Z","abstract_excerpt":"We study the topology of Vietoris--Rips complexes of finite grids on the torus. Let $T_{n,n}$ be the grid of $n\\times n$ points on the flat torus $S^1\\times S^1$, equipped with the $l^1$ metric. Let $\\mathrm{VR}(T_{n,n};k)$ be the Vietoris--Rips simplicial complex of this torus grid at scale $k\\ge 0$. For $n\\ge 7$ and small scales $2\\le k\\le \\frac{n-1}{3}$, the complex $\\mathrm{VR}(T_{n,n};k)$ is homotopy equivalent to the torus. For large scales $k\\ge 2\\lfloor\\frac{n}{2}\\rfloor$, the complex $\\mathrm{VR}(T_{n,n};k)$ is a simplex and hence contractible. Interesting topology arises over interme"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2502.07134","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2502.07134/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}