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Some ‘Converses’ to Intrinsic Linking Theorems

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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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Notes

  1. We are grateful to Florian Frick for allowing us to present this proof based on an idea he suggested. Alternative (earlier) proofs are presented in §3 and in [4, § 4–5].

  2. The condition \(\ell <k\) is present in [11, Lem. 1.1] but is not used in the proof.

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Authors and Affiliations

  1. Institute for Information Transmission Problems, Moscow, Russia

    Roman Karasev

  2. Moscow Institute of Physics and Technology, Independent University of Moscow, Moscow, Russia

    Arkadiy Skopenkov

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  1. Roman Karasev
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We are grateful to Florian Frick, Timur Garaev, and the anonymous referees for helpful discussions.

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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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