2017/05/22 by Eri Matsudo, Matsudo, Eri
Mathematics · #57M25 #FOS: Mathematics #Geometric Topology (math.GT) #math.GT #msc:57M25
paper · pdf · doi:10.48550/arxiv.1705.07567
15 pages, 25 figures v2:new main result added, v3:title changed with some minor corrections
arxiv created 2017/08/03 · arxiv updated 2017/08/04
The minimal coloring number of a ℤ-colorable link is the minimal number of colors for non-trivial ℤ-colorings on diagrams of the link. In this paper, we show that the minimal coloring number of any non-splittable ℤ-colorable links is four. As an example, we consider the link obtained by replacing each component of the given link with several parallel strands, which we call a parallel of a link. We show that an even parallel of a link is ℤ-colorable except for the case of 2 parallels with non-zero linking number. We then give a simple way to obtain a diagram which attains the minimal coloring number for such even parallels of links.