{
  "abstract": "For every torus knot, the rank of the differential of the Daemi-Scaduto S-complex of singular instanton homology is determined by the signature and the Alexander polynomial; this follows from theorems of Li-Ye and Daemi-Scaduto. For the torus knots $T(3,n)$ the differential has rank zero or one according to $n$ modulo $6$, and its rank equals Daemi and Scaduto's Frøyshov-type invariant; they proved this in two of the four residue classes and for $n=4,5$, and Hedden, Herald and Kirk computed it for $n\\le38$. Following Lewallen, Klassen and Hedden-Herald-Kirk, we describe the generators of the instanton complexes and their $\\mathbb{Z}/4$ gradings for two-bridge knots, whose irreducible traceless characters are all binary dihedral, and for $T(3,n)$, where at most one is. We also give a proof through Khovanov homology, using Schütz's computation for three-braids, of Li and Ye's computation of the rank of the instanton homology of $T(3,n)$, and compare the differential of the knot $8_{19}$ with the single pillowcase bigon of Hedden, Herald and Kirk.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.26095",
  "author_total": 1,
  "authors": [
    {
      "name": "Bernd Johannes Wuebben"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d94nstt7khmdd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.26095",
  "fulltext_url": null,
  "id": "d94nstt7khmdd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.26095",
  "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.SG"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7171861634",
  "pdf_url": "https://arxiv.org/pdf/2607.26095",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d94nstt7khmdd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2607.26095",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.26095"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Traceless $\\mathrm{SU}(2)$ characters and $\\mathbb{Z}/4$ instanton gradings for two-bridge and $(3,n)$-torus knots",
  "title_text": "Traceless SU(2) characters and ℤ/4 instanton gradings for two-bridge and (3,n)-torus knots",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/d94nstt7khmdd",
  "venue": "arXiv",
  "versions": []
}