Revision history [back]

The reason for p in p-colorability being required to be a prime number in knot theory is related to the fact that the fundamental group of a knot complement has a presentation in terms of p generators, where p is a prime number. This means that if a knot can be colored with p colors, then the group of all colorings has p generators, which is equivalent to saying that the fundamental group of the knot complement has p generators. If p is not a prime number, then the group of all colorings does not necessarily have p generators, and this can lead to complications in the analysis of the knot complement. Therefore, p is required to be a prime number in order to ensure that the coloring of the knot is related to the fundamental group of the knot complement.