{
  "abstract": "We study the $\\rm{PGL}_2$ skein algebras of small surfaces, focusing on the closed torus and the once-punctured torus. We provide explicit finite presentations by generators and relations for the $\\rm{PGL}_2$ skein algebra of the torus and once-punctured torus, analogous to those of Bullock and Przytycki for the Kauffman bracket skein algebras. We give explicit bases of the $\\rm{PGL}_2$ skein algebras of the torus and once-punctured torus and describe the multiplicative structure of the basis for the torus, analogous to the product-to-sum formula of Frohman and Gelca. We twist the basis for the torus to obtain one with structure constants in $\\mathbb{N}[q,q^{-1}].$ We compute the kernels of natural maps between the $\\rm{PGL}_2$ and Kauffman bracket skein algebras associated to the once-punctured torus and closed torus. We compute the center of the $\\rm{PGL}_2$ skein algebra of the torus, which depends on the specialization of the quantum parameter $q.$",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.01075",
  "author_total": 4,
  "authors": [
    {
      "name": "Cole Ellison"
    },
    {
      "name": "Vijay Higgins"
    },
    {
      "name": "Mark Sandey"
    },
    {
      "name": "Eden Sylvester"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6kexxsqkb2n7s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.01075",
  "fulltext_url": null,
  "id": "6kexxsqkb2n7s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.01075",
  "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"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7216044022",
  "pdf_url": "https://arxiv.org/pdf/2610.01075",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6kexxsqkb2n7s.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.01075",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.01075"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "$\\rm{PGL}_2$ skein algebras of small surfaces",
  "title_text": "PGL_2 skein algebras of small surfaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/6kexxsqkb2n7s",
  "venue": "arXiv",
  "versions": []
}