{
  "abstract": "Viro's Euler-integral polynomial $P_C(q)$ and the Lanzat--Polyak quantized-curvature polynomial $I_q(C)$ refine Arnold's invariants $J^-$ and $J^+$ for generic immersed one-component plane curves. We construct homological lifts of both. The bigraded region homology retains the singular homology of every connected Alexander-index region; its graded Euler characteristic is $P_C(q)$. The triply graded smoothing-circle homology is generated by the oriented circles of the orientation-preserving smoothing and decategorifies to the smoothing term in $I_q(C)$. Keeping the actual region summands and the boundary regions of every smoothing circle gives a homological refinement of the oriented smoothing configuration, or Seifert state. An infinite family proves strictness: both polynomial data and the ordinary homological lifts agree, while the component-graded region homology and the branch-decomposed circle homology distinguish every pair. Further constructions recover the full $I_q(C)$ by a vertex complex, realize the local change of its curvature integral by edge homology, and give a canonical two-state homology for unoriented curves. Viro described his Euler-integral formula as an analogue of face state-sum formulas for quantum knot polynomials. Through the categorifications developed here, we obtain one concrete homological face-state-sum model realizing that analogy.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11427",
  "author_total": 1,
  "authors": [
    {
      "name": "Noboru Ito"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cnw80k8xdzfhn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11427",
  "fulltext_url": null,
  "id": "cnw80k8xdzfhn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11427",
  "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": "W7212230868",
  "pdf_url": "https://arxiv.org/pdf/2609.11427",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cnw80k8xdzfhn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.11427",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11427"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Homological lifts of Arnold invariants $J^-$ and $J^+$",
  "title_text": "Homological lifts of Arnold invariants J^- and J^+",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/cnw80k8xdzfhn",
  "venue": "arXiv",
  "versions": []
}