  Using Ahlfors functions, Grunsky maps and the Bell representation theorem, we
show that a certain subset of the rational maps of degree $n$ forms a trivial
bundle over the moduli space of non-degenerate $n$-connected domains with one
marked tangent vector with fiber the $n$-fold symmetric product of the circle.
A consequence is that the set of rational Ahlfors functions of degree $n$ forms
a closed embedded submanifold inside the space of rational maps of degree $n$.
As an application, we show the existence of rational Ahlfors functions with
non-positive residues, resolving a question left open in a previous paper by
the authors.
