发表机构
King Abdulaziz University; National Technical University of Athens（阿卜杜勒阿齐兹国王大学; 雅典国立技术大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文引入图解 tangloid 代数及其 braidoid 子代数，通过扩展范畴和约化关系构造双线性配对，为 knotoid 等带端点图解结构提供代数框架。
AI 中文摘要
我们引入 \\(\emph{tangloid 代数}\\) \\(\mathcal{T}_n\\)，这是一个由扩展 tangloid 范畴 \\(\Ti\\) 产生的图解代数，及其子代数 \\(\emph{braidoid 代数}\\) \\(\mathcal{B}_n\\)。该构造始于无定向 tangloid 范畴 \\(UTC\\)，其灵感来自 Turaev 的 knotoid 理论，通过从无定向焊接 tangleoid 范畴 \\(UWTC\\) 中移除焊接关系，同时保留禁止移动并施加一些额外关系而获得。在此框架内，braidoid 范畴自然地作为 \\(UTC\\) 的子范畴出现。然后，我们通过附加一个占位态射 \\(\emptyset\\) 引入扩展 tangloid 范畴 \\(\Ti\\)，这使得生成元可以扩展为固定 \\(n\\) 盒的自同态。占位态射与交叉态射的相互作用产生了对角态射 \\(/\\) 和 \\(\backslash\\)，它们用于 \\(\Ti\\) 的定义关系。相应的扩展 braidoid 范畴 \\(\mathbf{Brd}_{\Ti}\\) 被实现为 \\(\Ti\\) 的子范畴。对于每个 \\(n\geq 0\\)，我们通过线性化自同态代数 \\(\operatorname{End}_{\Ti}(n)\\) 定义 tangloid 代数 \\(\mathcal{T}_n\\)，作为由图解生成元和关系呈现的含单位 \\(\mathbb{C}\\) 代数。我们还定义了 braidoid 子代数 \\(\mathcal{B}_n\\)，它为 braidoid 理论提供了代数框架。我们进一步通过施加关于平凡结和平凡 knotoid 分量的两个额外约化关系引入约化 tangloid 代数 \\(\delta\mathcal{T}_n\\)，并利用图解反射、复合和闭包在 \\(\delta\mathcal{T}_n\\) 上构造一个自然的图解双线性配对。我们的范畴和代数设置为研究具有内在端点的图解结构（即 knotoid、linkoid 和 braidoid）及相关图解代数提供了理论框架。
英文摘要
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.
