A coloring of knotted surface diagrams by a quandle is defined (cf. [CKS00])
in an analogous manner as for (classical) knots and links.
We specify a given orientation of a knotted surface diagram
by orientation normal vectors to broken sheets.
A coloring
for a knotted surface diagram by a quandle
is an assignment of an element (called a color) of a quandle
to each broken sheet such that
holds at every double point,
where
(or
, respectively)
is the color of the under-sheet
behind (or in front of) the over-sheet
with the color
,
where the normal of the over-sheet points from the under-sheet behind
it to the front.
The pair
is called
the source colors or ordered pair of colors of the double point curve.
The coloring rule is depicted in the left of Fig.
.
In the right of Fig.
, the situation of a coloring near a triple
point is depicted.
Note that a coloring of the ``bottom'' sheet that is divided into four broken sheet
looks exactly like a coloring of a classical knot diagram with region colors.
In particular, well-definedness of a coloring at a triple point
requires the self-distributivity.
The triple
of colors at a triple point
is called the source colors or ordered triple of colors
at a triple point.
In arguments similar to the classical case, it is seen by checking the Roseman
moves that there is a one-to-one correspondence between
the set of colorings
of a diagram of a surface
and that of a diagram
,
, obtained from
by
a Roseman move, and therefore, for any
and
representing the same
knotted surface.
In particular, again, the number of colorings is a knotted surface invariant.
The region colors are defined in a completely similar manner as well.