{
  "abstract": "We develop a filtered geometric framework for essential surfaces in knot exteriors. Let $γ$ be a unit-thickness representative of a knot type $K$ with $\\mathrm{Len}(γ)\\leΛ$, and let $F$ be a properly embedded essential surface in the exterior of a fixed-radius tube about $γ$, with $\\mathrm{Area}(F)\\leΔ$ and relative thickness at least $τ$, formulated through Federer reach and controlled boundary collars. We prove that this bounded-geometry pair space contains only finitely many pair-isotopy classes, and that equality of explicitly bounded canonical layered codes at resolution $\\varepsilon\\le c\\min\\{1,τ\\}$ implies ambient pair-isotopy. In a fixed exterior $E$, the surface systems visible in a geometric window form finite subcomplexes ${ES}_{Δ,τ}(E)$ that exhaust the essential-surface complex; isometries act levelwise and $C^{1,1}$ self-diffeomorphisms act with controlled reindexing. For a fixed two-sided surface, compressing disks are filtered in the same way, giving finite geometric witnesses for compressibility and weak reducibility and recovering the index-one characterization in Bachman's topological index theory. On the peripheral torus, every visible numerical boundary slope lies in an explicit writhe window, so that $|r|\\le C_{\\mathrm{BS}}Λ^{4/3}+w(Δ,τ)$. Together with finite Reidemeister certificates, the faithful codes give two independent finite recognition mechanisms, one for the knot and one for the carried essential-surface type. The framework is a smooth, triangulation-free analogue of the finiteness philosophy of normal surface theory; it does not assert that a knot exterior has only finitely many essential surfaces without geometric bounds.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.20844",
  "author_total": 1,
  "authors": [
    {
      "name": "Makoto Ozawa"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5jq6rc60can33.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.20844",
  "fulltext_url": null,
  "id": "5jq6rc60can33",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.20844",
  "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": "W7170770549",
  "pdf_url": "https://arxiv.org/pdf/2607.20844",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5jq6rc60can33.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2607.20844",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.20844"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Ropelength-Filtered Essential Surface Spaces",
  "title_text": "Ropelength-Filtered Essential Surface Spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/5jq6rc60can33",
  "venue": "arXiv",
  "versions": []
}