{
  "abstract": "The knot quandle, introduced independently by Joyce and Matveev, admits a geometric description in terms of nooses. Motivated by this description, we introduce the \\textit{loop fundamental quandle} $LQ(X,A,\\ast)$ of a pair $(X,A)$ with a fixed base point $\\ast\\in X$. Its elements are represented by pairs consisting of a loop in $A$ and a path joining the base point of the loop to the fixed base point $\\ast$. More generally, for a loop condition $P$ on $A$, invariant under free homotopy, we define the loop fundamental quandle $LQ(X,A,\\ast;P)$. We establish functoriality and homotopy invariance of the loop fundamental quandle and show that it admits a natural augmented quandle structure over $π_1(X,\\ast)$. For a knot $K$ in $S^3$, applying the construction to its exterior gives the \\textit{knot loop quandle} $LQ(K,\\ast;P)$. Appropriate choices of the loop condition recover the knot quandle, its $n$-th power quandles, and the full knot quandle. The construction is inspired by Joyce's noose-theoretic approach, but allows loop data that need not arise as the boundary of a disk satisfying the noose condition.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.16056",
  "author_total": 1,
  "authors": [
    {
      "name": "Seiichi Kamada"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ecyweppfbypb9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.16056",
  "fulltext_url": null,
  "id": "ecyweppfbypb9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.16056",
  "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": "W7221763462",
  "pdf_url": "https://arxiv.org/pdf/2610.16056",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ecyweppfbypb9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.16056",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.16056"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Loop Fundamental Quandles: A Geometric Generalization of the Knot Quandle",
  "title_text": "Loop Fundamental Quandles: A Geometric Generalization of the Knot Quandle",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/ecyweppfbypb9",
  "venue": "arXiv",
  "versions": []
}