  A low-dimensional version of our main result is the following `converse' of
the Conway-Gordon-Sachs Theorem on intrinsic linking of the graph $K_6$ in
3-space:
  For any integer $z$ there are 6 points $1,2,3,4,5,6$ in 3-space, of which
every two $i,j$ are joined by a polygonal line $ij$, the interior of one
polygonal line is disjoint with any other polygonal line, the linking
coefficient of any pair of disjoint 3-cycles except for $\{123,456\}$ is zero,
and for the exceptional pair $\{123,456\}$ is $2z+1$.
  We prove a higher-dimensional analogue, which is a `converse' of a lemma by
Segal-Spie\.z.
