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HEPTAGONAL KNOTS AND RADON PARTITIONS

  • Huh, Young-Sik (DEPARTMENT OF MATHEMATICS COLLEGE OF NATURAL SCIENCES HANYANG UNIVERSITY)
  • Received : 2009.10.26
  • Published : 2011.03.01

Abstract

We establish a necessary and sufficient condition for a heptagonal knot to be figure-8 knot. The condition is described by a set of Radon partitions formed by vertices of the heptagon. In addition we relate this result to the number of nontrivial heptagonal knots in linear embeddings of the complete graph $K_7$ into $\mathbb{R}^3$.

Acknowledgement

Supported by : National Research Foundation of Korea(NRF)

References

  1. J. L. Ramirez Alfonsin, Spatial graphs and oriented matroids: the trefoil, Discrete Comput. Geom. 22 (1999), no. 1, 149-158. https://doi.org/10.1007/PL00009446
  2. A. Bjorner, M. Las Vergnas, B. Sturmfels, N. White, and G. Ziegler, Oriented Matroids, Encyclopedia of Mathematics and its Applications, 46. Cambridge University Press, Cambridge, 1993.
  3. J. A. Calvo, Geometric Knot Theory, Ph. D. Thesis, Univ. Calif. Santa Barbara, 1998.
  4. J. H. Conway and McA. G. Gordon, Knots and links in spatial graphs, J. Graph Theory 7 (1983), no. 4, 445-453. https://doi.org/10.1002/jgt.3190070410
  5. L. Finschi, A Graph Theoretical Approach for Reconstruction and Generation of Oriented Matroids, Dissertation, Swiss Federal Institute of Technology (ETH) Zurich, 2001.
  6. L. Finschi, Homepage of oriented matroid, http://www.om.math.ethz.ch/
  7. E. Furstenberg, J. Li, and J. Schneider, Stick knots, Chaos Solitons Fractals 9 (1998), no. 4-5, 561-568. https://doi.org/10.1016/S0960-0779(97)00093-3
  8. Y. Huh and C. B. Jeon, Knots and links in linear embeddings of $K_{6}$, J. Korean Math. Soc. 44 (2007), no. 3, 661-671. https://doi.org/10.4134/JKMS.2007.44.3.661
  9. G. T. Jin, Polygon indices and superbridge indices of torus knots and links, J. Knot Theory Ramifications 6 (1997), no. 2, 281-289. https://doi.org/10.1142/S0218216597000170
  10. L. H. Kauffman, On Knots, Annals of Mathematics Studies, 115. Princeton University Press, Princeton, NJ, 1987.
  11. L. Mccabe, An upper bound on edge numbers of 2-bridge knots and links, J. Knot Theory Ramications 7 (1998), no. 6, 797-805. https://doi.org/10.1142/S0218216598000401
  12. K. Murasugi, Knot Theory and Its Applications, Translated from the 1993 Japanese original by Bohdan Kurpita. Birkhauser Boston, Inc., Boston, MA, 1996.
  13. S. Negami, Ramsey theorems for knots, links and spatial graphs, Trans. Amer. Math. Soc. 324 (1991), no. 2, 527-541. https://doi.org/10.2307/2001731
  14. R. Randell, An elementary invariant of knots, J. Knot Theory Ramifications 3 (1994), no. 3, 279-286. https://doi.org/10.1142/S0218216594000216
  15. R. Randell, Invariants of piecewise-linear knots, Knot theory (Warsaw, 1995), 307-319, Banach Center Publ., 42, Polish Acad. Sci., Warsaw, 1998.
  16. D. Rolfsen, Knots and links, Mathematics Lecture Series, No. 7. Publish or Perish, Inc., Berkeley, Calif., 1976.

Cited by

  1. KNOTTED HAMILTONIAN CYCLES IN LINEAR EMBEDDING OF K7 INTO ℝ3 vol.21, pp.14, 2012, https://doi.org/10.1142/S0218216512501325