On a Planarity Criterion Coming from Knot Theory

Abstract

A graph G is called "minimalizable" if a diagram with minimal crossing number can be obtained from an arbitrary diagram of G by crossing changes. If, furthermore, the minimal diagram is unique up to crossing changes then G is called "strongly minimalizable". In this article, it is explained how minimalizability of a graph is related to its automorphism group and it is shown that a graph is strongly minimalizable if the automorphism group is trivial or isomorphic to a product of symmetric groups. Then, the treatment of crossing number problems in graph theory by knot theoretical means is discussed and, as an example, a planarity criterion for minimalizable graphs is given.