{
  "abstract": "Takemura conjectured that the Conway polynomials $\\sum a_\\ell z^\\ell$ of the Turk's head link $Th(p,q)$ and $\\sum b_\\ell z^\\ell$ of $Th(q,p)$ satisfy $a_\\ell=b_\\ell$ for $\\ell\\equiv0,1\\pmod 4$ and $a_\\ell=-b_\\ell$ for $\\ell\\equiv2,3\\pmod4$, and proved it for $q=2,3$. We prove it for all $p,q\\ge1$, knots and links. Equivalently, $\\nabla_{Th(q,p)}(z)=c\\,\\nabla_{Th(p,q)}(iz)$ with $c=1$ if $\\gcd(p,q)$ is odd and $c=-i$ if it is even. The proof combines a product formula $\\nabla_{Th(p,q)}(z)^2=\\prod_{j=1}^{p-1}\\prod_{k=1}^{q-1}(z^2+2\\cos\\frac{2πk}{q}-2\\cos\\frac{2πj}{p})$, which comes from the reduced Burau representation, with the leading coefficients, read off the Seifert surface of the closed braid.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.32269",
  "author_total": 1,
  "authors": [
    {
      "name": "Werner Alois Stanggassinger"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9hhgb8x3hjt06.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.32269",
  "fulltext_url": null,
  "id": "9hhgb8x3hjt06",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.32269",
  "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": "W7223685215",
  "pdf_url": "https://arxiv.org/pdf/2610.32269",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9hhgb8x3hjt06.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2610.32269",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.32269"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Conway polynomials of $Th(p,q)$ and $Th(q,p)$: a proof of Takemura's conjecture",
  "title_text": "Conway polynomials of Th(p,q) and Th(q,p): a proof of Takemura's conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/9hhgb8x3hjt06",
  "venue": "arXiv",
  "versions": []
}