{
  "abstract": "The Melvin--Morton--Rozansky theorem relates the large-colour semiclassical limit of the coloured Jones polynomial to the inverse Alexander polynomial. Bar-Natan and Garoufalidis [BNG96] proved the theorem using weight systems and, more generally, established the corresponding formula for arbitrary complex semisimple Lie algebras. We give a direct proof of this semisimple Lie algebra generalization by analysing a state sum. For a complex semisimple Lie algebra $\\mathfrak{g}$ and a dominant integral weight $λ$, we use the embedding $V_{dλ}\\hookrightarrow V_λ^{\\otimes d}$ to analyse the state sum. Under the specialization $q=e^{h/d}$, we show that, as $d\\to\\infty$, only the identity terms and those proportional to $E_α\\otimes F_α$ contribute at leading order. It follows that the leading-order contribution decomposes over the positive roots, with each root contribution evaluated as in the $\\mathfrak{sl}_{2}$ case. Writing $t_α=e^{-2(λ,α)h}$, the limit is therefore given by the product of $Δ_{K}(t_α)^{-1}$ over the positive roots $α$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.10269",
  "author_total": 1,
  "authors": [
    {
      "name": "Atsuhide Nagasaka"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dcx0755ejdmzf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.10269",
  "fulltext_url": null,
  "id": "dcx0755ejdmzf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.10269",
  "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"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7221029102",
  "pdf_url": "https://arxiv.org/pdf/2610.10269",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dcx0755ejdmzf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2610.10269",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.10269"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A Semiclassical Limit of Large-Colour State Sums for Semisimple Lie Algebras",
  "title_text": "A Semiclassical Limit of Large-Colour State Sums for Semisimple Lie Algebras",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/dcx0755ejdmzf",
  "venue": "arXiv",
  "versions": []
}