Fox n-coloring

In the mathematical field of knot theory, Fox n-coloring is a method of specifying a representation of a knot group (or a link group) onto the dihedral group of order n where n is an odd integer by coloring arcs in a link diagram (the representation itself is also often called a Fox n-coloring). Ralph Fox discovered this method (and the special case of tricolorability) "in an effort to make the subject accessible to everyone" when he was explaining knot theory to undergraduate students at Haverford College in 1956. Fox n-coloring is an example of a conjugation quandle.

Let L be a link, and let π be the fundamental group of its complement. A representation ${\displaystyle \rho }$ of π onto ${\displaystyle D_{2n}}$ the dihedral group of order 2n is called a Fox n-coloring (or simply an n-coloring) of L. A link L which admits such a representation is said to be n-colorable, and ${\displaystyle \rho }$ is called an n-coloring of L. Such representations of link groups had been considered in the context of covering spaces since Reidemeister in 1929. [Actually, Reidemeister fully explained all this in 1926, on page 18 of "Knoten und Gruppen" in Hamburger Abhandlungen 5.]

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