Minimal coloring number on minimal diagrams for $\mathbb{Z}$colorable links
Abstract
It was shown that any $\mathbb{Z}$colorable link has a diagram which admits a nontrivial $\mathbb{Z}$coloring with at most four colors. In this paper, we consider minimal numbers of colors for nontrivial $\mathbb{Z}$colorings on minimal diagrams of $\mathbb{Z}$colorable links. We show, for any positive integer $N$, there exists a minimal diagram of a $\mathbb{Z}$colorable link such that any $\mathbb{Z}$coloring on the diagram has at least $N$ colors. On the other hand, it is shown that certain $\mathbb{Z}$colorable torus links have minimal diagrams admitting $\mathbb{Z}$colorings with only four colors.
 Publication:

arXiv eprints
 Pub Date:
 October 2017
 DOI:
 10.48550/arXiv.1710.03919
 arXiv:
 arXiv:1710.03919
 Bibcode:
 2017arXiv171003919I
 Keywords:

 Mathematics  Geometric Topology
 EPrint:
 7 pages, 9 figures