Special Issue: Proceedings of the Special Session “Low-Dimensional Topology and Its Relationships with Physics”, held at the AMS/EMS/SPM meeting in Porto, Portugal, June 10–13, 2015; Guest Editors: L. H. Kauffman and P. LopesNo Access

Minimal sufficient sets of colors and minimum number of colors

Jun Ge

Department of Mathematics, Sichuan Normal University, Chengdu 610066, Sichuan, P. R. China

E-mail Address: mathsgejun@163.com

Corresponding author.

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Xian’an Jin

School of Mathematical Sciences, Xiamen University, Xiamen 361005, Fujian, P. R. China

E-mail Address: xajin@xmu.edu.cn

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Louis H. Kauffman

Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851 S. Morgan St., Chicago, IL 60607-7045, USA

E-mail Address: kauffman@uic.edu

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Pedro Lopes

Center for Mathematical Analysis, Geometry, and Dynamical Systems, Department of Mathematics, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisbon, Portugal

E-mail Address: pelopes@math.tecnico.ulisboa.pt

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, and

Lianzhu Zhang

School of Mathematical Sciences, Xiamen University, Xiamen 361005, Fujian, P. R. China

E-mail Address: zhanglz@xmu.edu.cn

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https://doi.org/10.1142/S0218216517430088Cited by:1 (Source: Crossref)

Abstract

In this paper, we first investigate the minimal sufficient sets of colors for $p = 11$ $p = 11$ and $13$ $13$ . For odd prime $p$ $p$ and any $p$ $p$ -colorable link $L$ $L$ with $det L \neq 0$ $det L \neq 0$ , we give alternative proofs of ${mincol}_{p} L \geq 5$ ${mincol}_{p} L \geq 5$ for $p \geq 11$ $p \geq 11$ and ${mincol}_{p} L \geq 6$ ${mincol}_{p} L \geq 6$ for $p \geq 17$ $p \geq 17$ . We also 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 $11$ $11$ , and only one such class of five colors modulo 13.

Keywords:

AMSC: 57M27

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