{
  "abstract": "Let $J_r^{SU(n)}(K;q)$ denote the reduced $SU(n)$ quantum invariant of a zero-framed knot $K$, colored by the $r$th symmetric power of the defining representation and normalized to be $1$ for the unknot. For every fixed $n\\ge2$ we prove the Chen--Liu--Zhu cyclotomic expansion conjecture: there are unique coefficients $H_k^{(n)}(K;q)\\in\\mathbb{Z}[q^{\\pm1}]$ such that \\[ J_r^{SU(n)}(K;q)=\\sum_{k=0}^{r} \\left(\\prod_{i=0}^{k-1}\\{r-i\\}\\{r+n+i\\}\\right) H_k^{(n)}(K;q), \\] where $\\{m\\}=q^m-q^{-m}$. The finite dual interpolation formula of Beliakova--Gorsky gives an integral one-sided factorial expansion. After identifying their reduced scalar with the Habiro--Lê convention, we restrict the completed center to one-row colors. Completed Harish--Chandra reflection then yields inversion symmetry in the variable $z$, and integral descent through $X=z+z^{-1}$ converts the one-sided expansion into the two-sided Newton basis. Cyclotomic-local interpolation and a UFD denominator-removal argument prove Laurent integrality of the Newton coefficients. We also determine a natural coefficient ring for a rank-uniform expansion. For every zero-framed knot there are unique Laurent differential coefficients $G_k(K;A,q)\\in\\mathbb{Z}[A^{\\pm1},q^{\\pm1}]$. The associated Newton coefficients are Laurent polynomials in $A$ over $\\mathbb{Q}(q)$ whose values at every geometric node $A=q^n$, $n\\ge2$, lie in $\\mathbb{Z}[q^{\\pm1}]$. They define a two-variable Newton inverse-limit element whose positive-rank specializations recover all symmetric-color HOMFLY--PT polynomials. The completion is taken in the Newton kernels rather than coefficientwise at roots of unity. After a positive rank and a color have been fixed, the series is finite and may be evaluated at a root of unity.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29585",
  "author_total": 2,
  "authors": [
    {
      "name": "Honghuai Fang"
    },
    {
      "name": "Tian Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bm07z15qe935c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29585",
  "fulltext_url": null,
  "id": "bm07z15qe935c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29585",
  "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"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7204959230",
  "pdf_url": "https://arxiv.org/pdf/2608.29585",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bm07z15qe935c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.29585",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29585"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.QA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Cyclotomic Newton Expansions and a Rank-Uniform Integer-Valued Newton Completion",
  "title_text": "Cyclotomic Newton Expansions and a Rank-Uniform Integer-Valued Newton Completion",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/bm07z15qe935c",
  "venue": "arXiv",
  "versions": []
}