A generalisation of the Burau representation and groups $G_{n}^{3}$ for classical braids

TL;DR

This paper introduces a modified group related to braid dynamics, defines a new representation, and demonstrates its ability to detect the kernel of the classical Burau representation, advancing understanding of braid group representations.

Contribution

It presents a novel modification of the group G_n^3 and a new representation that enhances the analysis of braid group properties.

Findings

The new representation can detect the kernel of the Burau representation.

The modified group G_n^3 generalizes previous models of braid dynamics.

The approach provides a tool for studying classical braid group representations.

Abstract

We consider a certain modification of the group $G^3_n$ which describes dynamics of point configurations, in particular braids, and define a representation of the modified $G^3_n$. The braid representation induced is powerful enough to detect the kernel of the Burau representation.