{
  "abstract": "For p, q >= 3 with gcd(p,q) = 1 we show that the symmetry group of the Turk's head knot Th(p,q), the closure of the braid (sigma_1 sigma_2^{-1} sigma_3 sigma_4^{-1} ... sigma_{p-1}^{+-1})^q, is the dihedral group of order 4q. This proves Conjecture R of Di Prisa and Şavk for q >= 3; the case p = 3 is due to Sakuma and Weeks. The argument is diagrammatic: by the flyping theorem of Menasco and Thistlethwaite, the symmetries of a knot with a prime, reduced alternating diagram without non-trivial flypes are the symmetries of the diagram (we write out a proof for the diagrams at hand), and we show that the standard diagram of Th(p,q) has no non-trivial flypes and exactly 4q symmetries. The chirality of Th(p,q) for even p, which also follows from a theorem of Murasugi, is visible in the group. We compare the argument with the geometric one of Sakuma and Weeks for p = 3.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.16600",
  "author_total": 1,
  "authors": [
    {
      "name": "Werner Alois Stanggassinger"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7hhhgzvggwghg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.16600",
  "fulltext_url": null,
  "id": "7hhhgzvggwghg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.16600",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GT"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7221828371",
  "pdf_url": "https://arxiv.org/pdf/2610.16600",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7hhhgzvggwghg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2610.16600",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.16600"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The symmetry group of Turk's head knots",
  "title_text": "The symmetry group of Turk's head knots",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/7hhhgzvggwghg",
  "venue": "arXiv",
  "versions": []
}