{
  "abstract": "A twisted torus knot T(p,q;r,s) is obtained from the flat-torus diagram of T(p,q) by inserting s full twists along a disk meeting r adjacent strands. Our previous work determined its Alexander polynomial for 1<r<=max{|p|,|q|}. In this paper, we compute it in the complementary range max{|p|,|q|}<r<=|p|+|q| by placing the knot on a standard genus-two Heegaard surface and applying Fox calculus. We then combine the two calculations into a unified Alexander-polynomial formula valid for 1<r<=|p|+|q|. In the complementary range, we prove monicity and give an explicit formula for the breadth. This determines the Seifert genus for nonnegative twists and gives a genus lower bound for negative twists. We also exhibit infinite two-parameter families of non-L-space knots in the complementary range.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.34813",
  "author_total": 2,
  "authors": [
    {
      "name": "Adnan"
    },
    {
      "name": "Kyungbae Park"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5mrm31wdvrn5j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.34813",
  "fulltext_url": null,
  "id": "5mrm31wdvrn5j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.34813",
  "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": "W7224007737",
  "pdf_url": "https://arxiv.org/pdf/2610.34813",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5mrm31wdvrn5j.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.34813",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.34813"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The Alexander polynomial of twisted torus knots in the complementary range",
  "title_text": "The Alexander polynomial of twisted torus knots in the complementary range",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/5mrm31wdvrn5j",
  "venue": "arXiv",
  "versions": []
}