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Generalization of the Conway–Gordon Theorem and Intrinsic Linking on Complete Graphs

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Abstract

Conway and Gordon proved that for every spatial complete graph on six vertices, the sum of the linking numbers over all of the constituent two-component links is odd, and Kazakov and Korablev proved that for every spatial complete graph with arbitrary number of vertices greater than six, the sum of the linking numbers over all of the constituent two-component Hamiltonian links is even. In this paper, we show that for every spatial complete graph whose number of vertices is greater than six, the sum of the square of the linking numbers over all of the two-component Hamiltonian links is determined explicitly in terms of the sum over all of the triangle–triangle constituent links. As an application, we show that if the number of vertices is sufficiently large then every spatial complete graph contains a two-component Hamiltonian link whose absolute value of the linking number is arbitrary large. Some applications to rectilinear spatial complete graphs are also given.

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Acknowledgements

The authors are grateful to Professors Jae Choon Cha and Ayumu Inoue for their valuable comments.

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Authors and Affiliations

  1. Division of Mathematics, Graduate School of Science, Tokyo Woman’s Christian University, 2-6-1 Zempukuji, Suginami-ku, Tokyo, 167-8585, Japan

    Hiroko Morishita & Ryo Nikkuni

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  1. Hiroko Morishita
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  2. Ryo Nikkuni
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Correspondence to Ryo Nikkuni.

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Communicated by Kolja Knauer

This article is dedicated to Professor Yoshiyuki Ohyama on his 60th birthday

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The second author was supported by JSPS KAKENHI Grant nos. JP15K04881 and JP19K03500.

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Morishita, H., Nikkuni, R. Generalization of the Conway–Gordon Theorem and Intrinsic Linking on Complete Graphs. Ann. Comb. 25, 439–470 (2021). https://doi.org/10.1007/s00026-021-00536-5

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