Structure and Representations of a Diagrammatic Non-Commutative Tangloid Algebra

We introduce the \emph{tangloid algebra} ${\mathcal{T}}_n$, a diagrammatic algebra arising from the extended tangloid category $\text{Ti}$, 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 $\text{Ti}$ by adjoining a placeholder morphism $\mathrm{\varnothing }$, 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, $\mathrm{/}$ and $\mathrm{\backslash }$, which are used in the defining relations of $\text{Ti}$. The corresponding extended braidoid category ${\mathbf{B}\mathbf{r}\mathbf{d}}_{\text{Ti}}$ is realised as a subcategory of $\text{Ti}$. For each $n\ge 0$, we then define the tangloid algebra ${\mathcal{T}}_n$ by linearising the endomorphism algebra ${\mathrm{End}}_{\text{Ti}}(n)$, 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 $\delta {\mathcal{T}}_n$ by imposing two additional reduction relations for trivial knot and trivial knotoid components, and we construct a natural diagrammatic $\mathit{\text{bilinear}}\mathit{\text{pairing}}$ on $\delta {\mathcal{T}}_n$ 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.