My overarching question is why in the Artin presentation of the (geometrically defined) braid group there are no more than two relations. I've learned one way to prove this is using the fact that the group $Br_n$ is the $\pi_1$ of configuration space 
$Conf_n=(\mathbb{C}^n\backslash \{z_i=z_j, i\neq j\})/S_n$
together with the fact that when computing the $\pi_1$ of a CW complex you can restrict your attention to its 2-skeleton. 

Now I'm wondering if there is an explicit description of a cell decomposition for the configuration space (in particular of the  1- and 2-cells in it).