  We study the projection $\pi: M_d \to B_d$ which sends an affine conjugacy
class of polynomial $f: \mathbb{C}\to\mathbb{C}$ to the holomorphic conjugacy
class of the restriction of $f$ to its basin of infinity. When $B_d$ is
equipped with a dynamically natural Gromov-Hausdorff topology, the map $\pi$
becomes continuous and a homeomorphism on the shift locus. Our main result is
that all fibers of $\pi$ are connected. Consequently, quasiconformal and
topological basin-of-infinity conjugacy classes are also connected. The key
ingredient in the proof is an analysis of model surfaces and model maps,
branched covers between translation surfaces which model the local behavior of
a polynomial.
