Fox N-coloring - Generalization To G-coloring

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 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 A₅, T(3,5) has non-constant G-colorings.