Exactly fourteen intrinsically knotted graphs have 21 edges

Abstract

Throughout the article we will take an embedded graph to mean a graph embedded in R . We call a graph G intrinsically knotted if every embedding of the graph contains a knotted cycle. Conway and Gordon [2] showed that K7 , the complete graph with seven vertices, is an intrinsically knotted graph. A graph H is minor of another graph G if it can be obtained from G by contracting or deleting some edges. An intrinsically knotted graph is minor minimal intrinsically knotted provided no proper minor is intrinsically knotted. Robertson and Seymour [9] proved that there are only finite minor minimal intrinsically knotted graphs, but finding the complete set of them is still an open problem. However, it is well known that K7 and the thirteen graphs obtained from this graph by rY moves are minor minimal intrinsically knotted; see Conway and Gordon [2], and Kohara and Suzuki [6].