# Forks, Noodles and the Burau representation for $n=4$

## TL;DR

This paper investigates the faithfulness of the Burau representation for four-strand braids using homological techniques, proposing a conjecture and providing computational and geometric evidence.

## Contribution

It introduces a new approach using forks and noodles homological techniques to analyze the Burau representation at n=4, including a conjecture on its faithfulness.

## Key findings

- Proposes a conjecture implying faithfulness for n=4
- Provides geometric examples and computational data
- Offers arguments supporting the conjecture's validity

## Abstract

\begin{abstract} The reduced Burau representation is a natural action of the braid group $B_n$ on the first homology group $H_1({\tilde{D}}_n;\mathbb{Z})$ of a suitable infinite cyclic covering space ${\tilde{D}}_n$ of the $n$--punctured disc $D_n$. It is known that the Burau representation is faithful for $n\le 3$ and that it is not faithful for $n\ge 5$. We use forks and noodles homological techniques and Bokut--Vesnin generators to analyze the problem for $n=4$. We present a Conjecture implying faithfulness and a Lemma explaining the implication. We give some arguments suggesting why we expect the Conjecture to be true. Also, we give some geometrically calculated examples and information about data gathered using a C\texttt{++} program.