Involve: A Journal of Mathematics

  • Involve
  • Volume 10, Number 2 (2017), 291-316.

Fox coloring and the minimum number of colors

Mohamed Elhamdadi and Jeremy Kerr

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/involve.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We study Fox colorings of knots that are 13-colorable. We prove that any 13-colorable knot has a diagram that uses exactly five of the thirteen colors that are assigned to the arcs of the diagram. Due to an existing lower bound, this gives that the minimum number of colors of any 13-colorable knot is 5.

Article information

Source
Involve, Volume 10, Number 2 (2017), 291-316.

Dates
Received: 29 September 2015
Revised: 5 January 2016
Accepted: 24 January 2016
First available in Project Euclid: 13 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.involve/1513135632

Digital Object Identifier
doi:10.2140/involve.2017.10.291

Mathematical Reviews number (MathSciNet)
MR3574302

Zentralblatt MATH identifier
1358.57014

Subjects
Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

Keywords
knots fox colorings minimum number of colors

Citation

Elhamdadi, Mohamed; Kerr, Jeremy. Fox coloring and the minimum number of colors. Involve 10 (2017), no. 2, 291--316. doi:10.2140/involve.2017.10.291. https://projecteuclid.org/euclid.involve/1513135632


Export citation

References

  • F. Bento and P. Lopes, “The minimum number of Fox colors modulo 13 is 5”, preprint, 2015.
  • R. H. Fox, “A quick trip through knot theory”, pp. 120–167 in Topology of 3-manifolds and related topics (Athens, GA, 1961), Prentice-Hall, Englewood Cliffs, N.J., 1962.
  • C. Hayashi, M. Hayashi, and K. Oshiro, “On linear $n$-colorings for knots”, J. Knot Theory Ramifications 21:14 (2012), art. ID #1250123.
  • P. Lopes and J. Matias, “Minimum number of Fox colors for small primes”, J. Knot Theory Ramifications 21:3 (2012), art ID #1250025.
  • T. Nakamura, Y. Nakanishi, and S. Satoh, “The pallet graph of a Fox coloring”, Yokohama Math. J. 59 (2013), 91–97.
  • K. Oshiro, “Any 7-colorable knot can be colored by four colors”, J. Math. Soc. Japan 62:3 (2010), 963–973.
  • S. Satoh, “5-colored knot diagram with four colors”, Osaka J. Math. 46:4 (2009), 939–948.