that the Whitney trick can be made to work in dimension four in a certain limited class of problems (Theorem \[Whitney\]).

Theorem \[RST0\] above as well as the following Theorem \[RST\] are essentially due to Robertson, Seymour and Thomas [@RST], [@RST']; they had a different formulation but its equivalence with the present one is relatively easy [@M2] (see also Lemma \[panels\] and Remark \[erratum\] below).

\[RST\] Let $G$ be a graph.

\(a) Two linkless, knotless embeddings of $|G|$ in $S^3$ are inequivalent iff they differ already on $|H|$ for some subgraph $H$ of $G$, isomorphic to a subdivision of $K_5$ or $K_{3,3}$.

\(b) $|G|$ linklessly embeds in $S^3$ iff $G$ has no minor in the Petersen family.

The [*Petersen family*]{} of graphs is shown in Fig. 1 (disregard the colors for now) and includes the Petersen graph $P$, the complete graph $K_6$, the complete tripartite graph $K_{3,3,1}$, the graph $K_{4,4}\but$(edge), and three further graphs which we denote $\Gamma_7$, $\Gamma_8$ and $\Gamma_9$. These seven graphs can be defined as all the graphs obtainable from $K_6$ by a sequence of $\nabla{\mathrm Y}$- and ${\mathrm Y}\nabla$-[*exchanges*]{}, which as their name suggests interchange a copy of the $1$-skeleton of a $2$-simplex $\Delta^2\subset\Delta^3$ with a 