  A plane graph $H$ is a {\em plane minor} of a plane graph $G$ if there is a
sequence of vertex and edge deletions, and edge contractions performed on the
plane, that takes $G$ to $H$. Motivated by knot theory problems, it has been
asked if the plane minor relation is a well-quasi-order. We settle this in the
affirmative. We also prove an additional application to knot theory. If $L$ is
a link and $D$ is a link diagram, write $D\leadsto L$ if there is a sequence of
crossing exchanges and smoothings that takes $D$ to a diagram of $L$. We show
that, for each fixed link $L$, there is a polynomial-time algorithm that takes
as input a link diagram $D$ and answers whether or not $D\leadsto L$.
