{
  "abstract": "The knot invariants \\(V_n\\) arise from a rank-two Nichols algebra and a \\(4n\\)-dimensional right Yetter--Drinfeld module, whereas the colored Links--Gould invariants are defined from typical \\(U_q(\\mathfrak{sl}(2|1))\\)-modules. We prove that these two constructions agree, up to the mirror and parameter inversion forced by the right-module convention. The proof is structural. We first realize the braid operator \\(T_n\\) as a gauge transform of a canonical super Yetter--Drinfeld braiding. We then construct the relevant right--right paired double, prove that its Hopf pairing is perfect in every root degree, and obtain its completed universal \\(R\\)-matrix. An explicit Abelian Drinfeld twist separates this double into an all-odd \\(U_\\hbar(\\mathfrak{sl}(2|1))\\) root factor and a commutative central factor. Under this factorization, the Nichols module becomes a typical all-odd highest-weight module tensored with a one-dimensional central module. Finally, we transport duality, all four oriented crossings, partial transposes, and writhe normalization. For every oriented knot \\(\\mathcal K\\), every \\(n\\geq 1\\), and every \\(β\\neq 0,-1\\), the result is \\[ V_{n,\\mathcal K}\\!\\left(Q^{2nβ+n},Q^2\\right) = LG_{\\mathcal K}^{(n)}\\!\\left(Q^{-nβ},Q^{-1}\\right) = LG_{\\overline{\\mathcal K}}^{(n)}\\!\\left(Q^{nβ},Q\\right). \\] In particular, the scalar-identity property conjectured for the endomorphism-valued \\(V_n\\) construction follows from the simplicity of the corresponding typical module.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.28720",
  "author_total": 1,
  "authors": [
    {
      "name": "Jiuhe Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/27s2at7751dvh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.28720",
  "fulltext_url": null,
  "id": "27s2at7751dvh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.28720",
  "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.QA"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7204968375",
  "pdf_url": "https://arxiv.org/pdf/2608.28720",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/27s2at7751dvh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2608.28720",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.28720"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.QA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The \\(V_n\\) Invariants as Colored Links--Gould Invariants:A Root--Center Approach",
  "title_text": "The V_n Invariants as Colored Links--Gould Invariants:A Root--Center Approach",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/27s2at7751dvh",
  "venue": "arXiv",
  "versions": []
}