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.
Categories: math.GR