Some `converses' to intrinsic linking theorems

Abstract

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 joint by a polygonal line ij, the interior of one polygonal line is disjoint with any other polygonal line, the linking coefficient of any pair 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-Spież. * We would like to thank F. Frick for helpful discussions.