A KLR-like presentation for the bt-algebra

Abstract

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 \} $.