For a given knot diagram with a (Fox) -coloring (a coloring by ), its mirror has the -coloring (called the mirror coloring) that is the mirror image of . Specifically, let be the set of (over-)arcs of a diagram , then the mirror image has the set of arcs that is in a natural bijection with , such that each arc has its mirror , and vice-versa (for any arc , there is a unique arc such that ). Then is defined by .
In Fig. a -colored figure-eight knot diagram and its mirror with its mirror coloring are depicted. It is, then, natural to ask if an -colored knot diagram (a pair of a diagram and a coloring ) is equivalent to its mirror with its mirror coloring . If this is the case, we call the colored diagram amphicheiral (or achiral) with (respect to) the -coloring . Otherwise we say a diagram is chiral with (respect to) the -coloring.
The answer to the above question for the colored figure-eight knot is NO, and it is seen from the contributions of the -cocycle invariant with the dihedral quandle , as they are distinct between the left and right of Fig. . Many other achiral knots have colorings with respect to which they are chiral. On the other hand, there are many colorings of many knots that are achiral with the colorings.