  Using Thurston's characterization of postcritically finite rational functions
as branched coverings of the sphere to itself, we give a new method of
constructing new conformal dynamical systems out of old ones. Let $f(z)$ be a
rational map and suppose that the postcritical set $P(f)$ is finite. Let
$\alpha$ be an embedded closed arc in the sphere and suppose that $f|{\alpha}$
is a homeomorphism. Define a branched covering $g$ as follows. Cut the sphere
open along $\alpha$. Glue in a closed disc $D$. Map $S^{2} - \Int (D)$ via $f$
and $\Int (D)$ by a homeomorphism to the complement of $f(\alpha)$. We prove
theorems which give combinatorial conditions on $f$ and $\alpha$ for $g$ to be
equivalent in the sense of Thurston to a rational map. The main idea in our
proofs is a general theorem which forces a possible obstruction for $g$ away
from the disc $D$ on which the new dynamics is defined.
