{
  "abstract": "We introduce the \\emph{tangloid algebra} $\\mathcal{T}_n$, a diagrammatic algebra arising from the extended tangloid category $\\Ti$, and its subalgebra, the \\emph{braidoid algebra} $\\mathcal{B}_n$. The construction begins with the unoriented tangloid category $UTC$, inspired by Turaev's theory of knotoids, which is obtained from the unoriented welded tangleoid category $UWTC$ by removing the welded relation while retaining the forbidden moves and imposing some additional relations. Within this framework, the braidoid category arises naturally as a subcategory of $UTC$. We then introduce the extended tangloid category $\\Ti$ by adjoining a placeholder morphism $\\emptyset$, which allows the generators to be extended to endomorphisms of a fixed $n$-box. The interaction of the placeholder morphism with the crossing morphisms gives rise to diagonal morphisms, $/$ and $\\backslash$, which are used in the defining relations of $\\Ti$. The corresponding extended braidoid category $\\mathbf{Brd}_{\\Ti}$ is realised as a subcategory of $\\Ti$. For each $n\\geq 0$, we then define the tangloid algebra $\\mathcal{T}_n$ by linearising the endomorphism algebra $\\operatorname{End}_{\\Ti}(n)$, as a unital $\\mathbb{C}$--algebra presented by diagrammatic generators and relations. We also define the braidoid subalgebra $\\mathcal{B}_n$, which provides an algebraic framework for the theory of braidoids. We further introduce the reduced tangloid algebra $δ\\mathcal{T}_n$ by imposing two additional reduction relations for trivial knot and trivial knotoid components, and we construct a natural diagrammatic \\textit{bilinear pairing} on $δ\\mathcal{T}_n$ using diagrammatic reflection, composition, and closure. Our categorical and algebraic setting provides a theoretical framework for studying diagrammatic structures with intrinsic endpoints, namely knotoids, linkoids, and braidoids, and related diagram algebras.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.37524",
  "author_total": 2,
  "authors": [
    {
      "name": "Hadeel B. Albeladi"
    },
    {
      "name": "Sofia Lambropoulou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4eay33zbaa12k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.37524",
  "fulltext_url": null,
  "id": "4eay33zbaa12k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.37524",
  "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.AT"
    ],
    [
      "arxiv",
      "math.QA"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7214974337",
  "pdf_url": "https://arxiv.org/pdf/2609.37524",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4eay33zbaa12k.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.37524",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.37524"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.QA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Structure and Representations of a Diagrammatic Non-Commutative Tangloid Algebra",
  "title_text": "Structure and Representations of a Diagrammatic Non-Commutative Tangloid Algebra",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/4eay33zbaa12k",
  "venue": "arXiv",
  "versions": []
}