{
  "abstract": "This paper studies the swept-area cost of isotopies between thick knot representatives when the isotopy is required to stay inside a ropelength window: every intermediate curve has thickness at least one and length at most $Λ$. Without these constraints the swept area is the classical homotopy-area length on spaces of curves, and the induced distance on unparametrized curves is bounded below by the flat norm; see Yezzi--Mennucci and Michor--Mumford. We record the corresponding non-degeneracy on the ropelength-filtered moduli space, taken modulo orientation-preserving reparametrizations and Euclidean isometries, as a consequence of this classical lower bound. The ropelength window changes the theory in two ways. Distances are infinite between classes that are not yet connected at the level $Λ$, and all costs depend monotonically on the budget. We organize this dependence through budget--cost profiles and swept-area merge costs of admissible components. We prove their monotonicity and a transport estimate under whole-path simulations, and relate them to the merge scales of ropelength-filtered knot spaces. On the diagrammatic side, we construct a network with exact spatial endpoints whose path cost equals the infimal cost over diagrammatically generic isotopies, and show that the graph obtained by collapsing projection fibres gives only a lower bound, which can lose positive cost inside a fibre. We also give projected-area calibrations, exact formulas for concentric round and homothetic elliptical unknots, in which the window is not active, a labelled polygonal estimate, and a based loop-length structure on admissible fundamental groups. Existence of minimizing isotopies under the thickness and length constraints is left open.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.05557",
  "author_total": 1,
  "authors": [
    {
      "name": "Makoto Ozawa"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/84qz7rm5j9ekd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.05557",
  "fulltext_url": null,
  "id": "84qz7rm5j9ekd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.05557",
  "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"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7160631254",
  "pdf_url": "https://arxiv.org/pdf/2605.05557",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/84qz7rm5j9ekd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2605.05557",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.05557"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Ropelength-Filtered Swept-Area Geometry",
  "title_text": "Ropelength-Filtered Swept-Area Geometry",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/84qz7rm5j9ekd",
  "venue": "arXiv",
  "versions": []
}