Excluded minors and the ribbon graphs of knots

TL;DR

This paper studies the minor relations of ribbon graphs, focusing on those representing knot and link diagrams, and provides an excluded minor characterization for this family.

Contribution

It introduces an excluded minor characterization for ribbon graphs that correspond to knot and link diagrams, expanding the understanding of their structural properties.

Findings

Identifies the differences between ribbon graph minors and graph minors.

Provides an excluded minor characterization for ribbon graphs of knots and links.

Enhances the structural understanding of knot diagram representations.

Abstract

In this paper we consider minors of ribbon graphs (or, equivalently, cellularly embedded graphs). The theory of minors of ribbon graphs differs from that of graphs in that contracting loops is necessary and doing this can create additional vertices and components. Thus the ribbon graph minor relation is incompatible with the graph minor relation. We discuss excluded minor characterisations of minor closed families of ribbon graphs. Our main result is an excluded minor characterisation of the family of ribbon graphs that represent knot and link diagrams.