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 group of a link (not to be confused with 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.
Definition.
Let "L" be a link, and let formula_1 be the fundamental group of its complement. A representation formula_2 of formula_1 onto formula_4 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 formula_2 is called an "n"-coloring of "L". Such representations of groups of links 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.]
The group of a link is generated by paths from a basepoint in formula_6 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 formula_7 of the dihedral group, where "t" is a reflection and "s" is a generating (formula_8) rotation of the "n"-gon. The generators of the group of a link given above are in bijective correspondence with arcs of a link diagram, and if a generator maps to formula_9 we color the corresponding arc formula_10. This is called a Fox "n"-coloring of the link diagram, and it satisfies the following properties:
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 formula_2. By a theorem of Montesinos and Hilden, any closed oriented 3-manifold may be obtained this way for some knot "K" and formula_2 some tricoloring of "K". This is no longer true when "n" is greater than three.
Number of colorings.
The number of distinct Fox "n"-colorings of a link "L", denoted 
is an invariant of the link, which is easy to calculate by hand on any link diagram by coloring arcs according to the coloring rules. When counting colorings, by convention we also consider the case where all arcs are given the same color, and call such a coloring trivial. 
For example, the standard minimal crossing diagram of the Trefoil knot has 9 distinct tricolorings as seen in the figure:
The set of Fox 'n'-colorings of a link forms an abelian group formula_16, where the sum of two "n"-colorings is the "n"-coloring obtained by strandwise addition. This group splits as a direct sum
where the first summand corresponds to the "n" trivial (constant) colors, and nonzero elements of formula_18 summand correspond to nontrivial "n"-colorings ("modulo" translations obtained by adding a constant to each strand). 
If formula_19 is the connected sum operator and formula_20 and formula_21 are links, then
Generalization to "G"-coloring.
Let "L" be a link, and let "π" be the fundamental group of its complement, and let "G" be a group. A homomorphism formula_2 of "π" to "G" is called a "G"-coloring of "L". 
A "G"-coloring of a knot diagram is an induced assigning an element of "G" to the strands of "L" such that, at each crossing, if "c" is the element of "G" assigned to the overcrossing strand and if "a" and "b" are the elements of "G" assigned to the two undercrossing strands, then "a = c−1 b c" or "b = c−1 a c", depending on the orientation of the overcrossing strand. If the group "G" is dihedral of order "2n", this diagrammatic representation of a "G"-coloring reduces to a Fox "n"-coloring. The torus knot T(3,5) has only constant "n"-colorings, but for the group "G" equal to the alternating group "A"5, T(3,5) has non-constant "G"-colorings.