Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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COLORING LINKS BY THE SYMMETRIC GROUP OF DEGREE THREE

  • Kazuhiro Ichihara (Department of Mathematics College of Humanities and Sciences Nihon University) ;
  • Eri Matsudo (The Institute of Natural Sciences Nihon University)
  • Received : 2022.09.17
  • Accepted : 2023.03.02
  • Published : 2023.07.31
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Abstract

We consider the number of colors for colorings of links by the symmetric group S3 of degree 3. For knots, such a coloring corresponds to a Fox 3-coloring, and thus the number of colors must be 1 or 3. However, for links, there are colorings by S3 with 4 or 5 colors. In this paper, we show that if a 2-bridge link admits a coloring by S3 with 5 colors, then the link also admits such a coloring with only 4 colors.

Keywords

Acknowledgement

The authors would like to thank to Masaaki Suzuki for useful discussions. Also they thank to anonymous referee of the previous submission.

References

  1. R. H. Crowell and R. H. Fox, Introduction to Knot Theory, reprint of the 1963 original, Graduate Texts in Mathematics, No. 57, Springer, New York, 1977.
  2. L. H. Kauffman and P. M. Lopes, Determinants of rational knots, Discrete Math. Theor. Comput. Sci. 11 (2009), no. 2, 111-122.
  3. A. Kawauchi, A Survey of Knot Theory, translated and revised from the 1990 Japanese original by the author, Birkhauser Verlag, Basel, 1996.
  4. E. Matsudo, Minimal coloring number for Z-colorable links II, J. Knot Theory Ramifications 28 (2019), no. 7, 1950047, 7 pp. https://doi.org/10.1142/s0218216519500470
  5. K. Murasugi, Knot Theory and Its Applications, translated from the 1993 Japanese original by Bohdan Kurpita, Birkhauser Boston, Inc., Boston, MA, 1996.
  6. S. Satoh, 5-colored knot diagram with four colors, Osaka J. Math. 46 (2009), no. 4, 939-948. http://projecteuclid.org/euclid.ojm/1260892835
  7. M. Zhang, X. A. Jin, and Q. Y. Deng, The minimal coloring number of any non-splittable Z-colorable link is four, J. Knot Theory Ramifications 26 (2017), no. 13, 1750084, 18 pp. https://doi.org/10.1142/S0218216517500845