Colorings of Knotted Surface Diagrams by Quandles

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 $D$ by a quandle $X$ is an assignment of an element (called a color) of a quandle $X$ to each broken sheet such that $p*q=r$ holds at every double point, where $p$ (or $r$, respectively) is the color of the under-sheet behind (or in front of) the over-sheet with the color $q$, where the normal of the over-sheet points from the under-sheet behind it to the front. The pair $(p,q)\in X\times X$ 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 $(p,q,r) \in X\times X \times X$ of colors at a triple point is called the source colors or ordered triple of colors at a triple point.

Figure: Colors at double curves and $3$-cocycle 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 ${\rm Col}_X(D)$ of a diagram of a surface $K$ and that of a diagram $D'$, ${\rm Col}_X(D')$, obtained from $D$ by a Roseman move, and therefore, for any $D$ and $D'$ 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.