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28 March 2024
 
  » arxiv » 1501.2421

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Minimal sufficient sets of colors and minimum number of colors
Jun Ge ; Xian'an Jin ; Louis H. Kauffman ; Pedro Lopes ; Lianzhu Zhang ;
Date 11 Jan 2015
AbstractIn this paper we first investigate minimal sufficient sets of colors for p=11 and 13. For odd prime p and any p-colorable link L with non-zero determinant, we give alternative proofs of mincol_p L geq 5 for p geq 11 and mincol_p L geq 6 for p geq 17. We elaborate on equivalence classes of sets of distinct colors (on a given modulus) and prove that there are two such classes of five colors modulo 11, and only one such class of five colors modulo 13. Finally, we give a positive answer to a question raised by Nakamura, Nakanishi, and Satoh concerning an inequality involving crossing numbers. We show it is an equality only for the trefoil and for the figure-eight knots.
Source arXiv, 1501.2421
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