# Two-tone colorings and surjective dihedral representations for links

## TL;DR

This paper introduces a two-tone coloring method for links and establishes conditions under which link groups admit surjective dihedral representations, showing that links with three or more components can map onto dihedral groups of any degree.

## Contribution

It presents a novel two-tone coloring technique and proves that links with at least three components can have surjective homomorphisms to dihedral groups of arbitrary degree.

## Key findings

- Links with three or more components admit surjective dihedral representations of any degree.
- The two-tone coloring provides a new criterion for such surjective representations.
- The method extends the understanding of link group representations beyond knots.

## Abstract

It is well-known that a knot is Fox $n$-colorable for a prime $n$ if and only if the knot group admits a surjective homomorphism to the dihedral group of degree $n$. However, this is not the case for links with two or more components. In this paper, we introduce a two-tone coloring on a link diagram, and give a condition for links so that the link groups admit surjective representations to the dihedral groups. In particular, it is shown that the link group of any link with at least 3 components admits a surjective homomorphism to the dihedral group of arbitrary degree.