Tricolorability

In the mathematical field of knot theory, the tricolorability of a knot is the ability of a knot to be colored with three colors subject to certain rules. Tricolorability is an isotopy invariant, and hence can be used to distinguish between two different (non-isotopic) knots. In particular, since the unknot is not tricolorable, any tricolorable knot is necessarily nontrivial. In these rules a strand in a knot diagram will be a piece of the string that goes from one undercrossing to the next. A knot is tricolorable if each strand of the knot diagram can be colored one of three colors, subject to the following rules:

1. At least two colors must be used, and
2. At each crossing, the three incident strands are either all the same color or all different colors. Some references state instead that all three colors must be used. For a knot, this is equivalent to the definition above; however, for a link it is not. "The trefoil knot and trivial 2-link are tricolorable, but the unknot, Whitehead link, and figure-eight knot are not. If the projection of a knot is tricolorable, then Reidemeister moves on the knot preserve tricolorability, so either every projection of a knot is tricolorable or none is." Here is an example of how to color a knot in accordance of the rules of tricolorability. By convention, knot theorists use the colors red, green, and blue. The granny knot is tricolorable. In this coloring the three strands at every crossing have three different colors. Coloring one but not both of the trefoil knots all red would also give an admissible coloring. The true lover's knot is also tricolorable. Tricolorable knots with less than nine crossings include 61, 74, 77, 85, 810, 811, 815, 818, 819, 820, and 821. The figure-eight knot is not tricolorable. In the diagram shown, it has four strands with each pair of strands meeting at some crossing. If three of the strands had the same color, then all strands would be forced to be the same color. Otherwise each of these four strands must have a distinct color.