Suppose I have $n$ circles in $\mathbb{R}^2$ centered at $(\epsilon^i_x,\epsilon^i_y)$ infinitesimally close to the center, with radii $r_i=1+\epsilon^i_r$ infinitesimally close to 1. In general this will produce a picture like this
which has the following braiding pattern
has the pattern been studied studied for arbitrary placement and number of circles?