Excluded Minors and the Ribbon Graphs of Knots

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.