On intrinsically knotted or completely 3-linked graphs

Ryo Hanaki, Ryo Nikkuni, Kouki Taniyama, Akiko Yamazaki

Published 2010-06-03, updated 2011-01-21Version 4

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 any of whose 2-component sublink is nonsplittable. We show that a graph obtained from the complete graph on seven vertices by a finite sequence of $\triangle Y$-exchanges and $Y \triangle$-exchanges is a minor-minimal intrinsically knotted or completely 3-linked graph.