On intrinsically knotted or completely 3-linked graphs

TL;DR

This paper characterizes a class of graphs that are inherently knotted or linked in three-dimensional space, showing they are minor-minimal and obtained from the complete graph on seven vertices through specific transformations.

Contribution

It identifies a broad class of minor-minimal intrinsically knotted or 3-linked graphs generated by $ riangle Y$-exchanges and $Y riangle$-exchanges from K7.

Findings

Graphs obtained from K7 via $ riangle Y$-exchanges are minor-minimal intrinsically knotted or 3-linked.

The class includes all graphs with this property generated by these transformations.

The results provide a structural understanding of intrinsically knotted and 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 $\triangle Y$-exchanges and $Y \triangle$-exchanges is a minor-minimal intrinsically knotted or completely 3-linked graph.