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 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–Spież.
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Karasev, R., Skopenkov, A. Some ‘Converses’ to Intrinsic Linking Theorems. Discrete Comput Geom 70, 921–930 (2023). https://doi.org/10.1007/s00454-023-00505-0
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DOI: https://doi.org/10.1007/s00454-023-00505-0