On the subgroup of $B_4$ that contains the kernel of Burau representation

Abstract

It is known that there are braids $\alpha$ and $\beta$ in the braid group $B_4$, such that the group $⟨\alpha ,\beta ⟩$ is a fee subgroup \cite{7}, which contains the kernel $K$ of the Burau map ${\rho }_4:B_4\to GL\left(3,\mathbb{Z}[t,t^{-1}]\right)$ \cite{6}, \cite{4}. In this paper we will prove that $K$ is subgroup of $G=⟨\tau ,\Delta ⟩$, where $\tau$ and $\Delta$ are fourth and square roots of the generator $\theta$ of the center $Z$ of the group $B_4$. Consequently, we will write elements of $K$ in terms of ${\tau }^i,i=1,2,3$ and $\Delta$. Moreover, we will show that the quotient group $G/Z$ is isomorphic to the free product $Z_4\ast Z_2$.