Gr\"obner-Shirshov bases for braid groups in Adyan-Thurston generators

In this paper, we give a Gr"obner-Shirshov basis of the braid group $B_{n+1}$ in Adyan-Thurston generators. We also deal with the braid group of type ${\mathbf{B}}_{\mathbf{n}}$. As results, we obtain a new algorithm for getting the Adyan-Thurston normal form, and a new proof that the braid semigroup $B_{n+1}^{+}$ is the subsemigroup in $B_{n+1}$.