{
  "abstract": "The fact that every smooth knot type admits a Legendrian representative is a classical result in contact topology. However, the analogous surjectivity question was open at the parametric level. In this work we address the $n>1$ case. We prove that for every $n\\geq 3$, every knot type $\\mathcal K$, every Legendrian representative $\\mathcal L$ and every formal Legendrian representative $\\mathcal{FL}$, the associated group homomorphisms $π_n(\\mathcal{L})\\toπ_n(\\mathcal{K})$ and $π_n(\\mathcal{FL})\\toπ_n(\\mathcal{K})$ are never surjective. We then show that surjectivity at the $π_2$-level depends on the knot type. This work thus proves the presence of rigidity for parametric families at every higher homotopy level beyond $π_1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18492",
  "author_total": 1,
  "authors": [
    {
      "name": "Javier Martínez-Aguinaga"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1x30467jjn7rg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18492",
  "fulltext_url": null,
  "id": "1x30467jjn7rg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18492",
  "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.DG"
    ],
    [
      "arxiv",
      "math.GT"
    ],
    [
      "arxiv",
      "math.SG"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.18492",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1x30467jjn7rg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.18492",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18492"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Families of knots that cannot be made Legendrian parametrically",
  "title_text": "Families of knots that cannot be made Legendrian parametrically",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/1x30467jjn7rg",
  "venue": "arXiv",
  "versions": []
}