Journal of the Mathematical Society of Japan

Generalizations of the Conway–Gordon theorems and intrinsic knotting on complete graphs

Hiroko MORISHITA and Ryo NIKKUNI

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Abstract

In 1983, 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 that for every spatial complete graph on seven vertices, the sum of the Arf invariants over all of the Hamiltonian knots is odd. In 2009, the second author gave integral lifts of the Conway–Gordon theorems in terms of the square of the linking number and the second coefficient of the Conway polynomial. In this paper, we generalize the integral Conway–Gordon theorems to complete graphs with arbitrary number of vertices greater than or equal to six. As an application, we show that for every rectilinear spatial complete graph whose number of vertices is greater than or equal to six, the sum of the second coefficients of the Conway polynomials over all of the Hamiltonian knots is determined explicitly in terms of the number of triangle-triangle Hopf links.

Note

The second author was supported by JSPS KAKENHI Grant Number JP15K04881.

Article information

Source
J. Math. Soc. Japan, Advance publication (2019), 19 pages.

Dates
Received: 8 July 2018
First available in Project Euclid: 14 June 2019

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1560499222

Digital Object Identifier
doi:10.2969/jmsj/80858085

Subjects
Primary: 57M15: Relations with graph theory [See also 05Cxx]
Secondary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

Keywords
spatial graphs Conway–Gordon theorems

Citation

MORISHITA, Hiroko; NIKKUNI, Ryo. Generalizations of the Conway–Gordon theorems and intrinsic knotting on complete graphs. J. Math. Soc. Japan, advance publication, 14 June 2019. doi:10.2969/jmsj/80858085. https://projecteuclid.org/euclid.jmsj/1560499222


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References

  • [1] L. Abrams, B. Mellor and L. Trott, Counting links and knots in complete graphs, Tokyo J. Math., 36 (2013), 429–458.
  • [2] L. Abrams, B. Mellor and L. Trott, Gordian (Java computer program), available at http://myweb.lmu.edu/bmellor/research/Gordian.
  • [3] C. C. Adams, The knot book, An elementary introduction to the mathematical theory of knots, revised reprint of the 1994 original, Amer. Math. Soc., Providence, RI, 2004.
  • [4] C. C. Adams, B. M. Brennan, D. L. Greilsheimer and A. K. Woo, Stick numbers and composition of knots and links, J. Knot Theory Ramifications, 6 (1997), 149–161.
  • [5] P. Blain, G. Bowlin, J. Foisy, J. Hendricks and J. LaCombe, Knotted Hamiltonian cycles in spatial embeddings of complete graphs, New York J. Math., 13 (2007), 11–16.
  • [6] A. F. Brown, Embeddings of graphs in $E^{3}$, Ph. D. Dissertation, Kent State University, 1977.
  • [7] J. A. Calvo, Geometric knot spaces and polygonal isotopy, Knots in Hellas '98, Vol. 2 (Delphi), J. Knot Theory Ramifications, 10 (2001), 245–267.
  • [8] J. H. Conway and C. McA. Gordon, Knots and links in spatial graphs, J. Graph Theory, 7 (1983), 445–453.
  • [9] T. Endo and T. Otsuki, Notes on spatial representations of graphs, Hokkaido Math. J., 23 (1994), 383–398.
  • [10] N. J. Fine, Binomial coefficients modulo a prime, Amer. Math. Monthly, 54 (1947), 589–592.
  • [11] E. Flapan, T. Mattman, B. Mellor, R. Naimi and R. Nikkuni, Recent developments in spatial graph theory, Knots, links, spatial graphs, and algebraic invariants, Contemp. Math., 689, Amer. Math. Soc., Providence, RI, 2017, 81–102.
  • [12] T. Fleming and B. Mellor, Counting links in complete graphs, Osaka J. Math., 46 (2009), 173–201.
  • [13] J. Foisy, Corrigendum to: “Knotted Hamiltonian cycles in spatial embeddings of complete graphs” by P. Blain, G. Bowlin, J. Foisy, J. Hendricks and J. LaCombe, New York J. Math., 14 (2008), 285–287.
  • [14] 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.
  • [15] Y. Hirano, Knotted Hamiltonian cycles in spatial embeddings of complete graphs, Doctoral Thesis, Niigata University, 2010.
  • [16] Y. Hirano, Improved lower bound for the number of knotted Hamiltonian cycles in spatial embeddings of complete graphs, J. Knot Theory Ramifications, 19 (2010), 705–708.
  • [17] C. Hughes, Linked triangle pairs in a straight edge embedding of $K_6$, Pi Mu Epsilon J., 12 (2006), 213–218.
  • [18] Y. Huh and C. Jeon, Knots and links in linear embeddings of $K_6$, J. Korean Math. Soc., 44 (2007), 661–671.
  • [19] Y. Huh, Knotted Hamiltonian cycles in linear embedding of $K_7$ into ${\mathbb R}^{3}$, J. Knot Theory Ramifications, 21 (2012), 1250132, 14 pp.
  • [20] 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, http://www.f.waseda.jp/taniyama/SG2010/talks/19-7Jeon.pdf.
  • [21] L. H. Kauffman, Formal knot theory, Mathematical Notes, 30, Princeton University Press, Princeton, NJ, 1983.
  • [22] L. Ludwig and P. Arbisi, Linking in straight-edge embeddings of $K_7$, J. Knot Theory Ramifications, 19 (2010), 1431–1447.
  • [23] S. Negami, Ramsey theorems for knots, links and spatial graphs, Trans. Amer. Math. Soc., 324 (1991), 527–541.
  • [24] R. Nikkuni, A refinement of the Conway–Gordon theorems, Topology Appl., 156 (2009), 2782–2794.
  • [25] D. O'Donnol, Knotting and linking in the Petersen family, Osaka J. Math., 52 (2015), 1079–1100.
  • [26] T. Otsuki, Knots and links in certain spatial complete graphs, J. Combin. Theory Ser. B, 68 (1996), 23–35.
  • [27] M. Polyak and O. Viro, On the Casson knot invariant, Knots in Hellas '98, Vol. 3 (Delphi), J. Knot Theory Ramifications, 10 (2001), 711–738.
  • [28] J. L. Ramírez Alfonsín, Spatial graphs and oriented matroids: the trefoil, Discrete Comput. Geom., 22 (1999), 149–158.
  • [29] J. L. Ramírez Alfonsín, Spatial graphs, knots and the cyclic polytope, Beiträge Algebra Geom., 49 (2008), 301–314.
  • [30] D. Rolfsen, Knots and links, Mathematics Lecture Series, 7, Publish or Perish, Inc., Berkeley, CA, 1976.