Communications of the
Korean Mathematical Society CKMS

We consider the number of colors for 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.

Keywords: Link, coloring, symmetric group of degree 3

MSC numbers: 57K10

Supported by: The first author is partially supported by Grant-in-Aids for Scientific Research (C) (No. 18K03287), The Ministry of Education, Culture, Sports, Science and Technology, Japan.