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 K6 in 3-space: For any integer z there are six 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 number 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.