Some `converses' to intrinsic linking theorems

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.