in $\mathcal{A}$. All such detours are clearly homotopic to each other. A negative detour is defined similarly with $\ii$ replaced by $- \ii$.

Note that, for $v \in \mathcal{M}(e,r)$ and $s$ a reflection in $G(e,e,r)$, if there exists a positive detour from the base point $\underline{b} = (b_1,\dots,b_r)$ to $w.\underline{b}$, then it provides a braided reflection around the hyperplane attached to $s$.

The elements $i^*(\tau_k)$ are now easy-to-describe braided reflections, as the positive detours from $\underline{b}$ to their images by the corresponding reflections. In case $e = 2$, the given monoid is then clearly the classical Artin monoid of type $D_r$, so we can assume $e \geq 3$. The paths corresponding to $\xi_0^{2}$ and to its translates are homotopic to a line segment in $\mathcal{M}(e,r)$. The fact that $\tau'_1$ is a braided reflection essentially amounts to the fact that $i^*(g_0^{-1} .\tau_1)$ is a braided reflection in $\pi_1(\mathcal{M}(e,r)/G(e,e,r), g_0^{-1} . y)$, and this holds true because $\tau_1$ is a braided reflection in $\mathcal{M}^{\#}(e,r)/G(e,e,r)$.

We consider the plane $P$ defined by the equations $z_i = b_i$ for $i = 3,\dots,r$, and identify it to $\C^2$ through $(z_1,z_2)$. We let $P^0 = \C^2 \setminus \bigcup \{ z_2 = z_1 \eta \ | \ 