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.