{
  "abstract": "We investigate the expectation $E(v_2)$ of the Casson invariant $v_2$ for knots represented by two similar types of braids, the rosette $s$-braids which are products of $k$ factors $\\prod_{1\\leq 2i\\leq s}σ_{2i}^{\\pm 1}\\prod_{1\\leq 2i+\\leq s}σ_{2i+1}^{\\pm 1}$ and the quasitoric $s$-braids which are products of $k$ factors $σ_{1}^{\\pm 1}σ_{2}^{\\pm 1}...σ_{s-1}^{\\pm 1}$. If $k$ is of the form $ns+1$, $\\dfrac{E(v_2)}{c}\\sim \\dfrac{s+1}{24}$ where $c$ is the number of crossing points of braid; if $k$ is of the form $ns-1$, $\\dfrac{E(v_2)}{c}\\sim \\dfrac{5-s}{24}$.\\\\ If we go through all $3$-braids, regardless of the shape of their diagram, we get $\\dfrac{E(v_2)}{c}\\sim \\dfrac{1}{8}$. \\\\ For knots with a checkerboard diagram, $\\dfrac{E(v_2)}{c}\\sim \\dfrac{1}{16}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01188",
  "author_total": 1,
  "authors": [
    {
      "name": "Marina Ville"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/33q57yb16fmep.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01188",
  "fulltext_url": null,
  "id": "33q57yb16fmep",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01188",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.01188",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/33q57yb16fmep.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.01188",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01188"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Casson invariant of knots defined by random braids",
  "title_text": "Casson invariant of knots defined by random braids",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/33q57yb16fmep",
  "venue": "arXiv",
  "versions": []
}