{
  "abstract": "We report the first results of phenomenological analysis, based on extended set of explicitly calculated colored HOMFLY-PT polynomials in \\cite{knotebook}. They concern differential expansion (DE), which separates the dependencies on the knot ${\\cal K}$ and the representation $R$. %of colored HOMFLY-PT polynomials. DE is well understood for symmetric and rectangular representations, while for non-rectangular $R$ even the knot-independent basis $Z_R^X$ of DE is not known. We concentrate on two issues: the possible derivation of DE and on improved characteristic of its defect. We discuss how evolution families -- twist knots, double braids and 3-pretzels -- can be used to find DE by a peculiar triangular transformation of evolution coefficients. Once $Z^R_X$ are found for particular family, they are expected to stay basically the same for arbitrary knots -- modulo controllable reductions, characterized by the defect of DE, which puts knots into different universality classes. Contrary to original expectation, these classes appear labeled not by a single number $δ^{\\cal K}$, but by a whole non-decreasing sequence $\\{k_j\\}$, which is squeezed between two staircases, ${ entier}\\!\\left(\\frac{j-1}{δ+1}\\right)\\leq k_j\\leq{ entier}\\!\\left(\\frac{j-1}{{\\rm deg}\\,Δ}\\right)$, where the upper bound involves Alexander polynomial $Δ(q^2)$ and follows from the continuation of the Melvin--Morton--Rozansky-Kashaev limit to $A=q^{-m}$ with $m>0$, when the limit gives a {\\it positive} power of Alexander. The old assumption $δ+1={\\rm deg}\\,Δ$ is oversimplified and fails for 19 non-alternating knots with 11 and 12 crossings. Last, we list the differences of non-rectangularly colored HOMFLY for various mutant pairs to demonstrate their universality (knot-independence) and nice consistency with the spirit of DE.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.28531",
  "author_total": 2,
  "authors": [
    {
      "name": "Ya. Kononov"
    },
    {
      "name": "A. Morozov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7znfvcy1kv58t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.28531",
  "fulltext_url": null,
  "id": "7znfvcy1kv58t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.28531",
  "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"
    ],
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      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7223109653",
  "pdf_url": "https://arxiv.org/pdf/2610.28531",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7znfvcy1kv58t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:35Z",
      "native_id": "2610.28531",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.28531"
    }
  ],
  "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": "Towards differential expansion for generic representations",
  "title_text": "Towards differential expansion for generic representations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/7znfvcy1kv58t",
  "venue": "arXiv",
  "versions": []
}