A KLR-like presentation for the bt-algebra

We consider the bt-algebra ${\mathcal{E}}_n(q)$ of knot theory, defined over an arbitrary field $\mathbb{k}$. We find a KLR-like presentation for ${\mathcal{E}}_n(q)$ showing that it is a $\mathbb{Z}$-graded algebra if $q\in {\mathbb{k}}^{\times }\setminus \{1\}$ admits a square root in $\mathbb{k}$. We introduce the ordered bt-algebra ${\mathcal{E}}_n^{\mathrm{o}\mathrm{r}\mathrm{d}}(q)$ and show that it also has a KLR-like presentation, without restriction on $q\in {\mathbb{k}}^{\times }\setminus \{1\}$. In particular, ${\mathcal{E}}_n^{\mathrm{o}\mathrm{r}\mathrm{d}}(q)$ is a $\mathbb{Z}$-graded algebra for all $q\in {\mathbb{k}}^{\times }\setminus \{1\}$.