Structure and Representations of a Diagrammatic Non-Commutative Tangloid Algebra
Abstract
We introduce the \emph{tangloid algebra} 𝒯n, a diagrammatic algebra arising from the extended tangloid category Ti, and its subalgebra, the \emph{braidoid algebra} ℬn. 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 ∅, 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 BrdTi is realised as a subcategory of Ti. For each n≥ 0, we then define the tangloid algebra 𝒯n by linearising the endomorphism algebra EndTi(n), as a unital ℂ--algebra presented by diagrammatic generators and relations. We also define the braidoid subalgebra ℬn, which provides an algebraic framework for the theory of braidoids. We further introduce the reduced tangloid algebra δ𝒯n by imposing two additional reduction relations for trivial knot and trivial knotoid components, and we construct a natural diagrammatic \textit{bilinear pairing} on δ𝒯n 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.
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BibTeX
@misc{albeladi2026structure,
title = {{Structure and Representations of a Diagrammatic Non-Commutative Tangloid Algebra}},
author = {Hadeel B. Albeladi and Sofia Lambropoulou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37524},
eprint = {2609.37524},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37524},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Structure and Representations of a Diagrammatic Non-Commutative Tangloid Algebra
AU - Hadeel B. Albeladi
AU - Sofia Lambropoulou
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37524
UR - https://arxiv.org/abs/2609.37524
AB - We introduce the \emph{tangloid algebra} 𝒯_n, a diagrammatic algebra arising from the extended tangloid category Ti, and its subalgebra, the \emph{braidoid algebra} 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 ∅, 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 Brd_Ti is realised as a subcategory of Ti. For each n≥ 0, we then define the tangloid algebra 𝒯_n by linearising the endomorphism algebra End_Ti(n), as a unital ℂ--algebra presented by diagrammatic generators and relations. We also define the braidoid subalgebra ℬ_n, which provides an algebraic framework for the theory of braidoids. We further introduce the reduced tangloid algebra δ𝒯_n by imposing two additional reduction relations for trivial knot and trivial knotoid components, and we construct a natural diagrammatic \textit{bilinear pairing} on δ𝒯_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.
ER -