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 $\langle \alpha, \beta \rangle$ is a fee subgroup \cite{7}, which contains the kernel $K$ of the Burau map $\rho_4 : B_4 \to G L\left(3, \mathbb{Z}[t,t^{-1}]\right)$ \cite{6}, \cite{4}. In this paper we will prove that $K$ is subgroup of $G=\langle \tau, \Delta \rangle $, 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 $Z_4 *Z_2$.