{
  "abstract": "Each of the 12 known conjecturally optimal kissing configurations in dimensions 5, 6, and 7 carries a contact-graph polytope: a polytope whose vertices are the configuration points and whose edges are exactly its contacts. Eight of these polytopes are non-convex and distinct from their convex hulls, which include longer edges. Each polytope is radial about the sphere's center: its boundary meets every ray from the center exactly once. It is the unique radial polytope with those vertices and edges. Seven of the nine non-lattice configurations carry polytopes constructed from the lattice contact polytope in the same dimension by splitting some facets and reassembling the pieces. This split-and-reassemble construction describes the configurations by facets rather than by layers. The fold angles of the reassembled facets explain the inner products $1/(10-n)$, giving $1/5$, $1/4$, and $1/3$ in dimensions 5, 6, and 7, respectively. The contact graphs of all 12 configurations are infinitesimally rigid. We also prove that $Q_5$ and $R_5$ are infinitesimally jammed, extending the 10 previously established cases to all 12 configurations. All mathematical computations use exact arithmetic.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12640",
  "author_total": 1,
  "authors": [
    {
      "name": "Matthew Self"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4get65rfq1n1s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12640",
  "fulltext_url": null,
  "id": "4get65rfq1n1s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12640",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7212867160",
  "pdf_url": "https://arxiv.org/pdf/2609.12640",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4get65rfq1n1s.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2609.12640",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12640"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Non-convex polytopes realizing kissing contact graphs in dimensions 5-7",
  "title_text": "Non-convex polytopes realizing kissing contact graphs in dimensions 5-7",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/4get65rfq1n1s",
  "venue": "arXiv",
  "versions": []
}