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

TL;DR

This paper investigates the subgroup of the braid group B_4 containing the kernel of the Burau representation, revealing its structure and expressing its elements in terms of specific generators, and describing the quotient group structure.

Contribution

It proves that the kernel of the Burau map in B_4 is contained within a subgroup generated by specific roots of the center, and describes the quotient group structure as a free product.

Findings

Kernel of Burau map is contained in subgroup generated by τ and Δ.

Elements of the kernel can be expressed in terms of τ^i and Δ.

The quotient group G/Z is isomorphic to Z_4 * Z_2.

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 $Z_4 *Z_2$.