{
  "abstract": "HOMFLY-PT polynomials describe analytic continuation of Wilson averages in Chern-Simons theory to arbitrary complex values of two independent parameters $q$ and $A$. They also depend on the knot and representation of the gauge group. However, when $q^2$ is a primitive $m$-th root of unity, e.g. $q=\\pm e^{iπ/m}$, these polynomials exhibit a universal (knot-independent) factorization: $H_R$ becomes the product of the polynomials for the $m$-core of the Young diagram $R$ and of one and the same factor $H_{[m]}$ for every $m$-ribbon of $R$. In all the examples which we computed this factor is just the special polynomial at $A^m$. At $q=\\pm1$ factorization follows from the singularity of the cabled HOMFLY, which is the same for all representations. At $m>1$ the relevant singularity is in the projector, and what survives of it is a finite sum of braids $X_m$ -- the $m$-th power sum inside the cable, which can be freely moved through the other strands. For the proof we use the Murnaghan-Nakayama rule and the Adams operation. Discovery of this hidden structure explains the old factorization puzzle.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.08242",
  "author_total": 2,
  "authors": [
    {
      "name": "Ya. Kononov"
    },
    {
      "name": "A. Morozov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4yb2zcjna04t6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.08242",
  "fulltext_url": null,
  "id": "4yb2zcjna04t6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.08242",
  "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",
      "hep-th"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.GT"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7220984996",
  "pdf_url": "https://arxiv.org/pdf/2610.08242",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4yb2zcjna04t6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:22Z",
      "native_id": "2610.08242",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.08242"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:hep-th",
      "subfield": "high-energy"
    }
  ],
  "title": "What a knot sees at a root of unity",
  "title_text": "What a knot sees at a root of unity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/4yb2zcjna04t6",
  "venue": "arXiv",
  "versions": []
}