{
  "abstract": "We study the $d$-invariant from Heegaard Floer homology for Brieskorn spheres $Σ(p,q,r)$. Using surgeries on two singular fibers, we establish new relations of the $d$-invariants among various Brieskorn spheres. For $p=2$, we show that $d$ never vanishes if $q/r$ lies in certain intervals on the rational line. Altogether, we completely classify Brieskorn spheres $Σ(2,q,r)$ with $d=0$. The zero set is expressed using submonoids of $\\mathrm{GL}(2,\\mathbb{Z})$ and the Stern--Brocot tree. As an application, we prove that most Brieskorn spheres $Σ(2,q,r)$ do not bound a rational homology $4$-ball.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.36472",
  "author_total": 4,
  "authors": [
    {
      "name": "Shimal Harichurn"
    },
    {
      "name": "Isabella Khan"
    },
    {
      "name": "András Némethi"
    },
    {
      "name": "Josef Svoboda"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6gsyk03jfj1fz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36472",
  "fulltext_url": null,
  "id": "6gsyk03jfj1fz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.36472",
  "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.GT"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7224121743",
  "pdf_url": "https://arxiv.org/pdf/2610.36472",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6gsyk03jfj1fz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.36472",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36472"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "On the $d$-invariants of Brieskorn spheres and $\\mathrm{GL}(2,\\mathbb{Z})$",
  "title_text": "On the d-invariants of Brieskorn spheres and GL(2,ℤ)",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/6gsyk03jfj1fz",
  "venue": "arXiv",
  "versions": []
}