Definition
Let L be a link, and let π be the fundamental group of its complement. A representation of π onto 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 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.
The link group is generated paths from a basepoint in to the boundary of a tubular neighbourhood of the link, around a meridian of the tubular neighbourhood, and back to the basepoint. By surjectivity of the representation these generators must map to reflections of a regular n-gon. Such reflections correspond to elements of the dihedral group, where t is a reflection and s is a generating rotation of the n-gon. The generators of the link group given above are in bijective correspondence with arcs of a link diagram, and if a generator maps to we color the corresponding arc . This is called a Fox n-coloring of the link diagram, and it satisfies the following properties:
- At least two colors are used (by surjectivity of ).
- Around a crossing, the average of the colors of the undercrossing arcs equals the color of the overcrossing arc (because is a representation of the link group).
A n-colored link yields a 3-manifold M by taking the (irregular) dihedral covering of the 3-sphere branched over L with monodromy given by . By a theorem of Montesinos and Hilden, and closed oriented 3-manifold may be obtained this way for some knot K any some tricoloring of K. This is no longer true when n is greater than three.
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