Knotting and linking in the Petersen family

Abstract

This paper extends the work of Nikkuni [4] finding an explicit relationship for the graph $K_{3, 3, 1}$ between knotting and linking, which relates the sum of the squares of linking numbers of links in the embedding and the second coefficient of the Conway polynomial of certain cycles in the embedding. Then we use this and other similar relationships to better understand the relationship between knotting and linking in the Petersen family. The Petersen family is the set of minor minimal intrinsically linked graphs. We prove that if such a spatial graph is complexly algebraically linked then it is knotted.