vertices can only occur in dimensions $n=0,2,4,8,16$ and with a $\bmod 2$ cohomology ring isomorphic to that of the respective projective plane [@BK1] (see also [@La2]). In dimensions $2$ and $4$ these are unique (up to a relabelling of vertices); an algebraic topology proof can be found in [@ArM] and a combinatorial one in [@BD2]. In dimension $8$, there exist three self-dual subcomplexes of $\partial\Delta^{14}$, all triangulating a certain $8$-manifold $\Ham P^2_{(?)}$ [@BK2]. The existence in dimension $16$ seems to be still open.

Theorem \[self-dual\] now implies

\[1.10\] The join of $r$ copies of $\R P^2$, $c$ copies of $\C P^2$ and $h$ copies of $\Ham P^2_{(?)}$ embeds in $S^{5r+8c+14h-1}$ and does not embed in $S^{5r+8c+14h-2}$. Every embedding $g$ of this join in $S^{5r+8c+14h-1}$ is inequivalent to $hg$, where $h$ is an orientation reversing homeomorphism of the sphere.

We note that $K_6$ admits an embedding in $S^3$ that links any given disjoint circuits $|C|$, $|C'|$ in $|K_6|$ with any given odd linking number and does not link any other disjoint pair of circuits. Similar arguments work to prove the same assertion for any other graph of the Petersen family (cf. Remark \[hemi-exchanges\]).

Indeed, one of $C$, $C'$ bounds a simplex 