{
  "abstract": "Habiro's cyclotomic expansion of the colored Jones polynomial has a higher rank analogue conjectured by Chen--Liu--Zhu for colored $SU(n)$ invariants. We prove this conjecture for torus knots $T(2,2p+1)$ and give a Bailey theoretic realization of the cyclotomic coefficients. For $n\\geq2$ and $K_p=T(2,2p+1)$, the colored $SU(n)$ invariants admit an expansion $$ J_N^{SU(n)}(K_p;q) = \\sum_{m=0}^{N} \\left(\\prod_{j=0}^{m-1}\\{N-j\\}\\{N+n+j\\}\\right) H_m^{(n,p)}(q), $$ where $H_m^{(n,p)}(q)\\in\\mathbb Z[q^{\\pm1}]$ is independent of the color $N$. The proof identifies the cyclotomic basis with a Newton basis and rewrites the resulting Newton coefficients as an ordinary Bailey transform. The Lin--Zheng formula for $T(2,2p+1)$ then gives a well-poised Bailey kernel. A terminating very-well-poised ${}_6ϕ_5$ summation diagonalizes the kernel and reduces the integrality problem to an ordinary Bailey transition. We prove a uniform integrality theorem for these transitions in a formal integral $q$-difference operator algebra. As a direct corollary, the expansion yields the corresponding congruence relations and proves part(i) of the Chen--Liu--Zhu $SU(n)$ volume conjecture for $T(2,2p+1)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13766",
  "author_total": 1,
  "authors": [
    {
      "name": "Chuwen Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1jkag5b86rpvk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13766",
  "fulltext_url": null,
  "id": "1jkag5b86rpvk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13766",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GT"
    ],
    [
      "arxiv",
      "math.QA"
    ],
    [
      "arxiv",
      "math.RT"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7213338511",
  "pdf_url": "https://arxiv.org/pdf/2609.13766",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1jkag5b86rpvk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.13766",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13766"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.QA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Cyclotomic expansions of colored $SU(n)$ invariants of two-strand torus knots and Bailey transforms",
  "title_text": "Cyclotomic expansions of colored SU(n) invariants of two-strand torus knots and Bailey transforms",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/1jkag5b86rpvk",
  "venue": "arXiv",
  "versions": []
}