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


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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  1. 1.

    L. Abrams, B. Mellor and L. Trott, Counting links and knots in complete graphs, Tokyo J. Math. 36 (2013), 429–458.

    MathSciNet  Article  Google Scholar 

  2. 2.

    L. Abrams, B. Mellor and L. Trott, Gordian (Java computer program), available at

  3. 3.

    A. F. Brown, Embeddings of graphs in \(E^{3}\), Ph. D. Dissertation, Kent State University, 1977. 5 10 (2001), 245–267.

  4. 4.

    J. H. Conway and C. McA. Gordon, Knots and links in spatial graphs, J. Graph Theory 7 (1983), 445–453.

    MathSciNet  Article  Google Scholar 

  5. 5.

    T. Endo and T. Otsuki, Notes on spatial representations of graphs, Hokkaido Math. J. 23 (1994), 383–398.

    MathSciNet  Article  Google Scholar 

  6. 6.

    E. Flapan, Intrinsic knotting and linking of complete graphs, Algebr. Geom. Topol. 2 (2002), 371–380.

    MathSciNet  Article  Google Scholar 

  7. 7.

    E. Flapan and K. Kozai, Linking number and writhe in random linear embeddings of graphs, J. Math. Chem. 54 (2016), 1117–1133.

    MathSciNet  Article  Google Scholar 

  8. 8.

    E. Flapan, K. Kozai and R. Nikkuni, Stick number of non-paneled knotless spatial graphs, New York J. Math. 26 (2020), 836–852.

  9. 9.

    E. Flapan, T. Mattman, B. Mellor, R. Naimi and R. Nikkuni, Recent developments in spatial graph theory, Knots, links, spatial graphs, and algebraic invariants, 81–102, Contemp. Math., 689, Amer. Math. Soc., Providence, RI, 2017.

  10. 10.

    T. Fleming and B. Mellor, Counting links in complete graphs, Osaka J. Math. 46 (2009), 173–201.

    MathSciNet  MATH  Google Scholar 

  11. 11.

    H. Hashimoto and R. Nikkuni, Conway-Gordon type theorem for the complete four-partite graph \(K_{3,3,1,1}\), New York J. Math. 20 (2014), 471–495.

    MathSciNet  MATH  Google Scholar 

  12. 12.

    C. Hughes, Linked triangle pairs in a straight edge embedding of \(K_6\), Pi Mu Epsilon J. 12 (2006), 213–218.

    Google Scholar 

  13. 13.

    Y. Huh and C. Jeon, Knots and links in linear embeddings of \(K_6\), J. Korean Math. Soc. 44 (2007), 661–671.

    MathSciNet  Article  Google Scholar 

  14. 14.

    Y. Huh, Knotted Hamiltonian cycles in linear embedding of \(K_7\) into \({{\mathbb{R}}}^{3}\), J. Knot Theory Ramifications 21 (2012), 1250132, 14 pp.

  15. 15.

    C. B. Jeon, G. T. Jin, H. J. Lee, S. J. Park, H. J. Huh, J. W. Jung, W. S. Nam and M. S. Sim, Number of knots and links in linear \(K_7\), slides from the International Workshop on Spatial Graphs (2010),

  16. 16.

    A. A. Kazakov and Ph. G. Korablev, Triviality of the Conway-Gordon function \(\omega _{2}\) for spatial complete graphs, J. Math. Sci. (N.Y.) 203 (2014), 490–498.

    MathSciNet  Article  Google Scholar 

  17. 17.

    H. Morishita and R. Nikkuni, Generalizations of the Conway-Gordon theorems and intrinsic knotting on complete graphs, J. Math. Soc. Japan 71 (2019), 1223–1241.

    MathSciNet  Article  Google Scholar 

  18. 18.

    R. Naimi and E. Pavelescu, Linear embeddings of \(K_9\) are triple linked, J. Knot Theory Ramifications 23 (2014), 1420001, 9 pp.

  19. 19.

    R. Naimi and E. Pavelescu, On the number of links in a linearly embedded \(K_{3,3,1}\), J. Knot Theory Ramifications 24 (2015), 1550041, 21 pp.

  20. 20.

    R. Nikkuni, A refinement of the Conway-Gordon theorems, Topology Appl. 156 (2009), 2782–2794.

    MathSciNet  Article  Google Scholar 

  21. 21.

    T. Otsuki, Knots and links in certain spatial complete graphs, J. Combin. Theory Ser. B 68 (1996), 23–35.

    MathSciNet  Article  Google Scholar 

  22. 22.

    J. L. Ramírez Alfonsín, Spatial graphs and oriented matroids: the trefoil, Discrete Comput. Geom. 22 (1999), 149–158.

    MathSciNet  Article  Google Scholar 

  23. 23.

    M. Shirai and K. Taniyama, A large complete graph in a space contains a link with large link invariant, J. Knot Theory Ramifications 12 (2003), 915–919.

    MathSciNet  Article  Google Scholar 

  24. 24.

    A. Yu. Vesnin and A. V. Litvintseva, On linking of hamiltonian pairs of cycles in spatial graphs (in Russian), Sib. Èlektron. Mat. Izv. 7 (2010), 383–393

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

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Correspondence to Ryo Nikkuni.

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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.

Communicated by Kolja Knauer

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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).

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  • Spatial graphs
  • Conway–Gordon theorems

Mathematics Subject Classification

  • Primary 57M15
  • Secondary 57K10