Counting links and knots in complete graphs

Abstract

We investigate the minimal number of links and knots in embeddings of complete partite graphs in S3. We provide exact values or bounds on the minimal number of links for all complete partite graphs with all but 4 vertices in one partition, or with 9 vertices in total. In particular, we find that the minimal number of links in an embedding of K44,1 is 74. We also provide exact values or bounds on the minimal number of knots for all complete partite graphs with 8 vertices.