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 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 $△Y$-exchanges and $Y△$-exchanges is a minor-minimal intrinsically knotted or completely 3-linked graph.

17 pages, 9 figures

References in corpus (1)

• Graphs of 20 edges are 2-apex, hence unknotted

Cited by in corpus (9)

• Many, many more intrinsically knotted graphs
• Graphs on 21 edges that are not 2--apex
• $△Y$-exchanges and the Conway-Gordon theorems
• Maximal knotless graphs
• Intrinsically knotted graphs with 21 edges
• A new intrinsically knotted graph with 22 edges
• Conway-Gordon type theorem for the complete four-partite graph $K_{3,3,1,1}$
• On intrinsically knotted and linked graphs
• More intrinsically knotted graphs with 22 edges and the restoring method