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 $\Bbbk$. We find a KLR-like presentation for ${\mathcal E}_n(q)$ showing that it is a $\mathbb Z$-graded algebra if $q \in \Bbbk^{\times} \setminus \{1 \}$ admits a square root in $\Bbbk$. We introduce the ordered bt-algebra ${\mathcal E}^{\rm{ord}}_n(q)$ and show that it also has a KLR-like presentation, without restriction on $q \in \Bbbk^{\times} \setminus \{1 \}$. In particular, ${\mathcal E}^{\rm{ord}}_n(q)$ is a $\mathbb Z$-graded algebra for all $q \in \Bbbk^{\times} \setminus \{1 \}$.