Burau representation of $B_4$ and quantization of the rational projective plane

TL;DR

This paper explores the Burau representation of the braid group B4, classifies its orbits on rational projective planes, and investigates a q-deformation linking to q-rationals, revealing new algebraic structures.

Contribution

It introduces a classification of orbits under the Burau representation and establishes a connection between q-deformed projective lines and q-rationals.

Findings

Classification of B4-orbits on $ ext{P}^2( ext{Q})$

Existence of embeddings of $ ext{P}^1( ext{Q}(q))$ related to q-rationals

Analysis of B4-action on $ ext{P}^2( ext{Z}(q))$

Abstract

The braid group $B_4$ naturally acts on the rational projective plane $\mathbb{P}^2(\mathbb{Q})$, this action corresponds to the classical integral reduced Burau representation of $B_4$. The first result of this paper is a classification of the orbits of this action. The Burau representation then defines an action of $B_4$ on $\mathbb{P}^2(\mathbb{Z}(q))$, where $q$ is a formal parameter and $\mathbb{Z}(q)$ is the field of rational functions in $q$ with integer coefficients. We study orbits of the $B_4$-action on $\mathbb{P}^2(\mathbb{Z}(q))$, and show existence of embeddings of the $q$-deformed projective line $\mathbb{P}^1(\mathbb{Z}(q))$ that precisely correspond to the notion of $q$-rationals due to Morier-Genoud and Ovsienko.