{
  "abstract": "Kotelskiy, Watson and Zibrowius show that the Bar-Natan invariant $\\unicode{x0414}(T)$ associated to a $4$-ended oriented tangle $T$ extends to a twisted complex $\\unicode{x0414}^\\infty(T)$ over an $A_\\infty$-enhancement of the Bar-Natan algebra. We prove that $\\unicode{x0414}^\\infty(T)$ itself is a tangle invariant, settling a conjecture from their work. We deduce this from an invariance statement for a \\emph{matrix multifactorization} $\\mathcal M(T)$, generalizing a construction of Ballinger to the coefficient ring $\\mathbb Z[G]$. This multifactorization is well defined up to a notion of 1-homotopy equivalence. The proof rests on explicit formulas for special deformation retracts of Koszul matrix factorizations, which realize delooping in this setting and may be of independent interest.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.10262",
  "author_total": 1,
  "authors": [
    {
      "name": "Tomás Mejía-Gómez"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8dqbrmahyyyj4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.10262",
  "fulltext_url": null,
  "id": "8dqbrmahyyyj4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.10262",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.GT"
    ],
    [
      "arxiv",
      "math.QA"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7221011696",
  "pdf_url": "https://arxiv.org/pdf/2610.10262",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8dqbrmahyyyj4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2610.10262",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.10262"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.QA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "On the $A_\\infty$-extension of Bar-Natan multicurves",
  "title_text": "On the A_∞-extension of Bar-Natan multicurves",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/8dqbrmahyyyj4",
  "venue": "arXiv",
  "versions": []
}