On intrinsically knotted or completely 3-linked graphs

Abstract

We say that a graph is intrinsically knotted or completely 3-linked if every embedding of the graph into the 3-sphere contains a nontrivial knot or a 3-component link each of whose 2-component sublinks is nonsplittable. We show that a graph obtained from the complete graph on seven vertices by a finite sequence of ΔY-exchanges and YΔ-exchanges is a minor-minimal intrinsically knotted or completely 3-linked graph. © 2011 by Pacific Journal of Mathematics.