{
  "abstract": "We consider the three-strand pretzel knot $P_a=P(a,-a-2,-(a+1)^2/2)$ and prove that, when $a\\equiv1\\pmod4$, its Alexander polynomial fails the Fox--Milnor condition. The same remains true after adding any number of opposite odd parameter pairs $q,-q$ and reordering the parameters. Combined with work of Kim--Lee--Song, this gives the same conclusion for every odd $a\\ge3$. In particular, no knot in Lecuona's exceptional family is algebraically or topologically slice. Together with earlier work of Lecuona, Miller, and Kim--Lee--Song, this completes the slice--ribbon and topological-sliceness classifications for three-strand pretzel knots and removes the nonexceptional hypothesis from the Lecuona--Wand classification of prime fibered ribbon pretzel knots.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.34522",
  "author_total": 1,
  "authors": [
    {
      "name": "Suman Saurabh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5rx1bv1vwywwd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.34522",
  "fulltext_url": null,
  "id": "5rx1bv1vwywwd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.34522",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GT"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7214785908",
  "pdf_url": "https://arxiv.org/pdf/2609.34522",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5rx1bv1vwywwd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2609.34522",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.34522"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Exceptional pretzel knots are not algebraically slice",
  "title_text": "Exceptional pretzel knots are not algebraically slice",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/5rx1bv1vwywwd",
  "venue": "arXiv",
  "versions": []
}