  The space of monic squarefree complex polynomials has a stratification
according to the multiplicities of the critical points. We introduce a method
to study these strata by way of the infinite-area translation surface
associated to the logarithmic derivative $df/f$ of the polynomial. We determine
the monodromy of these strata in the braid group, thus describing which
braidings of the roots are possible if the orders of the critical points are
required to stay fixed. Mirroring the story for holomorphic differentials on
higher-genus surfaces, we find the answer is governed by the framing of the
punctured disk induced by the horizontal foliation on the translation surface.
