4$, $\hat K_7$ itself contains a [*proper*]{} subcomplex isomorphic to the $2$-skeleton of the $6$-simplex, and it is well-known (see [@M2]) that $|(\Delta^6)^{(2)}|$ still has a nonzero $\bmod 2$ van Kampen obstruction. The similar proper subcomplex of $\hat K_{3,3,1,1}$ is discussed in §\[dichotomial spheres\] below.

Let $E_n$ stand for “the problem of embeddability of a certain $n$-polyhedron in $S^{2n}$”, and let $L_n$ stand for “the problem of linkless embeddability of a certain $n$-polyhedron in $S^{2n+1}$”; it is understood that the polyhedron is specified, but nevertheless omitted in the notation. In §\[commensurability\] we describe a reduction (if and only if) of every $E_n$ to an $L_n$; and of every $L_n$ to an $E_{n+1}$ (Theorem \[commensuration\]). The construction is geometric, i.e. does not involve configuration spaces. The case $n=1$ was also done by van der Holst [@Ho].

Let us mention other known relations between embeddability and linkless embeddability. (i) M. Skopenkov has derived the non-embeddability of a certain $n$-polyhedron in $S^{2n}$ from non-existence of linkless embeddings of its links of vertices in $S^{2n-1}$ (thereby reducing a certain $E_n$ to a few $L_{n-1}$’s) by a geometric argument [@Sk-M2]. (ii) Conversely, the author derived the linkless non-embeddability of the $n$-skeleton of the $(2n+3)$-simplex in 