{
  "abstract": "Let $N$ denote the closed non-orientable surface of Euler characteristic $-1$, i.e. the connected sum of three real projective planes $N = \\mathbb{RP}^2 \\# \\mathbb{RP}^2 \\# \\mathbb{RP}^2$, with fundamental group presented as $\\langle A,B,C \\mid A^2 B^2 C^2 = I \\rangle$ via the standard symmetric cross-cap model. Building on the classical trace-coordinate methods of Fricke, Vogt, and Goldman for free groups, and on the qualitative study of representation varieties of non-orientable surfaces by Benyash-Krivets and Chernousov, we give an explicit affine equation for a dense open piece of the $\\mathrm{SL}_2(\\mathbb{C})$ GIT character variety $X(N)$. We prove that the trace coordinates $a = \\mathrm{tr}(A)$, $b = \\mathrm{tr}(B)$, $c = \\mathrm{tr}(C)$, $p = \\mathrm{tr}(AB)$ satisfy the polynomial relation $c^2 = abp - a^2 - b^2 + 4$, that this hypersurface is smooth and irreducible, and that on the open locus $c \\neq 0$ it is isomorphic to the image of $X(N)$ under the trace map, with the remaining trace coordinates $\\mathrm{tr}(BC)$, $\\mathrm{tr}(CA)$, $\\mathrm{tr}(ABC)$ recovered as regular functions on this locus. We further identify the boundary locus $c = 0$ with representations satisfying the matrix identity $A^2 B^2 = -I$, and describe its representation-theoretic structure, while leaving a complete global classification along this locus as an open problem. All key trace identities are proved from Cayley-Hamilton first principles and independently checked numerically against explicit matrices.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.08513",
  "author_total": 1,
  "authors": [
    {
      "name": "Yinxi Chen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9rjyv990146gc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.08513",
  "fulltext_url": null,
  "id": "9rjyv990146gc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.08513",
  "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.AG"
    ],
    [
      "arxiv",
      "math.AT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.08513",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9rjyv990146gc.ris",
  "sources": [
    {
      "first_seen": "2026-11-10T06:21:07Z",
      "native_id": "2611.08513",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.08513"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "An Explicit Equation for the $\\mathrm{SL}_2(\\mathbb{C})$ Character Variety of $\\mathbb{RP}^2 \\# \\mathbb{RP}^2 \\# \\mathbb{RP}^2$",
  "title_text": "An Explicit Equation for the SL_2(ℂ) Character Variety of ℝℙ^2 # ℝℙ^2 # ℝℙ^2",
  "type": "preprint",
  "updated_at": "2026-11-10T06:21:07Z",
  "url": "https://research.draconitic.com/paper/9rjyv990146gc",
  "venue": "arXiv",
  "versions": []
}