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

## TL;DR

This paper proves that the plane minor relation is a well-quasi-order and provides a polynomial-time algorithm to determine if a link diagram can be transformed into a fixed link via crossing exchanges and smoothings.

## Contribution

It establishes the well-quasi-order property for plane minors and introduces an efficient algorithm for link diagram transformations related to knot theory.

## Key findings

- Plane minor relation is a well-quasi-order.
- Polynomial-time algorithm for link diagram transformation.
- Application to knot theory problems.

## Abstract

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 $D\leadsto L$.