{"paper":{"title":"On Vietoris--Rips complexes (with scale 3) of hypercube graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.CO","authors_text":"Samir Shukla","submitted_at":"2022-02-06T11:02:47Z","abstract_excerpt":"For a metric space $(X, d)$ and a scale parameter $r \\geq 0$, the Vietoris-Rips complex $\\mathcal{VR}(X;r)$ is a simplicial complex on vertex set $X$, where a finite set $\\sigma \\subseteq X$ is a simplex if and only if diameter of $\\sigma$ is at most $r$. For $n \\geq 1$, let $\\mathbb{I}_n$ denotes the $n$-dimensional hypercube graph. In this paper, we show that $\\mathcal{VR}(\\mathbb{I}_n;r)$ has non trivial reduced homology only in dimensions $4$ and $7$. Therefore, we answer a question posed by Adamaszek and Adams recently.\n  A (finite) simplicial complex $\\Delta$ is $d$-collapsible if it can"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2202.02756","kind":"arxiv","version":3},"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/2202.02756/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"}