  The space of monic squarefree polynomials has a stratification according to
the multiplicities of the critical points, called the equicritical
stratification. Tracking the positions of roots and critical points, there is a
map from the fundamental group of a stratum into a braid group. We give a
complete determination of this map. It turns out to be characterized by the
geometry of the translation surface structure on $\mathbb{CP}^1$ induced by the
logarithmic derivative $df/f$ of a polynomial in the stratum.
