{
  "abstract": "Positive integer solutions to $x^2+y^2+z^2- x y z =D$ correspond to the important Markov (Markoff) numbers when $D=0$. From a given solution triple, three more are found with Vieta involutions, making an infinite tree of solutions. Starting instead with three real numbers greater than $2$ gives lengths of closed geodesics in a punctured torus. In this paper we study these tree structures for any real $D$. A continuous function, originally related to a norm on homology, encodes all the numbers on each of these trees. The properties of this function are developed here in general, with a self-contained exposition, showing that the usual $D=0$ case is part of a bigger picture. Encoding function graphs are shown to be convex for $D<4$, straight lines for $D=4$ and concave for $D>4$. Among other consequences are generalizations to all $D$ of: estimates for counting numbers on these trees, descriptions of the geometry of the corresponding lattice curves, uniqueness conditions, and identities of McShane and Hines.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14149",
  "author_total": 1,
  "authors": [
    {
      "name": "Cormac O'Sullivan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8qvka88ysshv7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14149",
  "fulltext_url": null,
  "id": "8qvka88ysshv7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14149",
  "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.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7213290105",
  "pdf_url": "https://arxiv.org/pdf/2609.14149",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8qvka88ysshv7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.14149",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14149"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Patterns in the Markov numbers and their generalizations",
  "title_text": "Patterns in the Markov numbers and their generalizations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/8qvka88ysshv7",
  "venue": "arXiv",
  "versions": []
}