{
  "abstract": "We prove the rectangular cyclotomic expansion conjecture of Kameyama, Nawata, Tao, and Zhang for reduced colored HOMFLY-PT polynomials of arbitrary zero-framed knots. We also prove their explicit double-twist formula for every rectangular color and all pairs of nonzero integer twist parameters. For each knot, we construct a double Newton expansion with coefficients in $\\mathbb Z[A^{\\pm1},q^{\\pm1}]$. These coefficients are uniquely determined and independent of the width and height of the rectangular color. The construction uses finite-rank representation symmetries and integral interpolation. The double-twist formula follows from a simultaneous exterior-power comparison of the two four-point Casimir operators and the known one-row formula. Finite transition formulas express the double Newton coefficients of double-twist knots in terms of the prescribed twist coefficients.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.00303",
  "author_total": 2,
  "authors": [
    {
      "name": "Honghuai Fang"
    },
    {
      "name": "Tian Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0yy5nv6nm7xg0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.00303",
  "fulltext_url": null,
  "id": "0yy5nv6nm7xg0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.00303",
  "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.GT"
    ],
    [
      "arxiv",
      "math.QA"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7215785483",
  "pdf_url": "https://arxiv.org/pdf/2610.00303",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0yy5nv6nm7xg0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.00303",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.00303"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.QA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Rectangular Cyclotomic Expansions of Colored HOMFLY-PT Polynomials",
  "title_text": "Rectangular Cyclotomic Expansions of Colored HOMFLY-PT Polynomials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/0yy5nv6nm7xg0",
  "venue": "arXiv",
  "versions": []
}