{
  "abstract": "We construct homological lifts of Arnold's strangeness $\\operatorname{St}$ for long and closed plane curves. For long curves, we introduce a new $q$-variable quantization polynomial $\\operatorname{St}_q$, an edge state sum, designed for categorification and construct a triply graded homology that recovers $\\operatorname{St}_q$. We prove strictness by exhibiting long curves with the same number of crossings and the same $\\operatorname{St}_q$ (in particular, the same rotation number and $\\operatorname{St}$) that are distinguished by the homology. For closed curves, we introduce a corresponding new polynomial $\\operatorname{St}_q$ and construct an edge homology. In this case $\\operatorname{St}_q$ is recovered from a signed $q$-character $\\operatorname{ch}_q$ defined on the canonical cycles of the homology rather than directly from a graded rank. Again, we exhibit closed curves with the same number of crossings and the same $\\operatorname{St}_q$ (in particular, the same rotation number and $\\operatorname{St}$) that are distinguished by the homology.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.36387",
  "author_total": 1,
  "authors": [
    {
      "name": "Noboru Ito"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dn9ped19fcdx4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36387",
  "fulltext_url": null,
  "id": "dn9ped19fcdx4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.36387",
  "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": "W7224134314",
  "pdf_url": "https://arxiv.org/pdf/2610.36387",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dn9ped19fcdx4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.36387",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36387"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Homological lifts of Arnold strangeness",
  "title_text": "Homological lifts of Arnold strangeness",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/dn9ped19fcdx4",
  "venue": "arXiv",
  "versions": []
}