{
  "abstract": "We use reflection and rotation signatures of dihedral groups to develop a Smith normal form method for computing the conjugation quandle colorings across all dihedral groups. Let $L = L_1 \\cup \\cdots \\cup L_\\ell$ be a link with link components $L_1, \\ldots, L_\\ell$, and let $\\operatorname{Conj}(D_n)$ be the conjugation quandle of the dihedral group $D_n$. We use the semidirect-product structure $D_n \\cong \\mathbb{Z}_n \\rtimes \\mathbb{Z}_2$ to separate the coloring into a component signature. This component signature tells us whether each link component is colored by rotations or reflections, and the exponents are assigned separately under modulus $n$. Once the component signature is determined, every crossing relation becomes an equation that determines an integer matrix $M_τ$. The abelian group $A_τ(L)$, which we call the signature module, is represented by an integer matrix $M_τ$. We prove that the free rank and nonunit Smith normal form entries, or equivalently the isomorphism class of $A_τ(L)$, are preserved under the Reidemeister moves. The complete family of signature modules also determines the entire family of counting invariants for all $D_n$. The all-reflection system is the same as the Fox-$n$ colorings, while the mixed rotation-reflection signatures contain information that gives a stronger invariant. We also show that each signature module is the $t_k = (-1)^{τ_k}$ specialization of the multivariable Alexander module, and extend the coloring interpretation of the module to generalized dihedral groups $\\operatorname{Dih}(B)$ for every abelian group $B$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13326",
  "author_total": 1,
  "authors": [
    {
      "name": "Zining Fan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c55bb7y7vvqk2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13326",
  "fulltext_url": null,
  "id": "c55bb7y7vvqk2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13326",
  "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.GT"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7213265054",
  "pdf_url": "https://arxiv.org/pdf/2609.13326",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c55bb7y7vvqk2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.13326",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13326"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Signature Modules and the Dihedral Conjugation-Quandle Counting Invariant",
  "title_text": "Signature Modules and the Dihedral Conjugation-Quandle Counting Invariant",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/c55bb7y7vvqk2",
  "venue": "arXiv",
  "versions": []
}