of a $1$-polyhedron $\Gamma$ in $S^m$ [*knotless*]{} if for every circuit $C\subset\Gamma$, the restriction of $g$ to $C$ is an unknot. We call an embedding $g$ of a polyhedron $X$ into $S^m$ [*linkless*]{} if for every two disjoint closed subpolyhedra of $g(X)$, one is contained in an $m$-ball disjoint from the other one. An $n$-polyhedron admitting no linkless embedding in $S^{2n+1}$ is also known in the literature (at least for $n=1$) as an “intrinsically linked” polyhedron.

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For $n>1$, the notion of a linkless embedding in $S^{2n+1}$ can be reformulated in terms of linking numbers of pairs of [*co*]{}cycles (see Lemma \[circuits\] and Lemma \[is-linkless\]). When $n=1$, variations of the notion of “linking” (such as existence of a nontrivial two-component sublink, or of a two-component sublink with nonzero linking number) lead to inequivalent versions of the notion of a linkless embedding in $\R^3$. It is amazing, however, that they all become equivalent upon adding a quantifier:

\[RST0\] Let $G$ be a graph. If $|G|$ admits an embedding in $S^3$ that links every pair of disjoint circuits with an even linking number, then $|G|$ admits a linkless, knotless embedding in $S^3$.

This is a very powerful result; it implies, [*inter alia*]{}, 