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

Well-quasi-order of plane minors and an application to link diagrams