Coloring links by the symmetric group of degree three

TL;DR

This paper investigates the coloring of links using the symmetric group S_3, revealing that certain 5-colorings in 2-bridge links can be reduced to 4-colorings, expanding understanding of link colorings.

Contribution

It demonstrates that for 2-bridge links, 5-colorings by S_3 can always be reduced to 4-colorings, providing new insights into link coloring properties.

Findings

2-bridge links with 5-colorings also admit 4-colorings

Colorings by S_3 differ between knots and links

The number of colors in link colorings can be reduced under certain conditions

Abstract

We consider the number of colors for the colorings of links by the symmetric group $S_3$ of degree $3$. For knots, such a coloring corresponds to a Fox 3-coloring, and thus the number of colors must be 1 or 3. However, for links, there are colorings by $S_3$ with 4 or 5 colors. In this paper, we show that if a 2-bridge link admits a coloring by $S_3$ with 5 colors, then the link also admits such a coloring with only 4 colors.