Colorings of regions

For a coloring ${\mathcal C}$ of a knot diagram $K$, there is a coloring of regions that extend ${\mathcal C}$ as depicted in Fig. . A coloring of regions is an assignment of quandle elements to regions that satisfies the following requirement: Suppose the two regions $R_1$ and $R_2$ are divided by an arc $\alpha$ such that the normal to $\alpha$ points from $R_1$ to $R_2$, and let $x$ be a color of $R_1$. Then the color of $R_2$ is required to be $x * {\mathcal C}(\alpha)$. At a crossing, there are four regions. There is a unique region among the four from which all the normals of the arcs point to other regions. Such a region is called the source region at a crossing.

Figure: Quandle colorings of regions

In Fig. , it is seen that colorings of regions (region colors) are well-defined. In the left of the figure, a positive crossing is depicted. There are two ways to go from the source region (leftmost region) to the rightmost region, through upper arcs and lower arcs. Through upper arcs we obtain the color $(x*y)*z$, and through the lower arcs $(x*z)*(y*z)$, that coincide by a quandle condition. The triple $(x,y,z)$ is called the ordered triple of colors at the crossing. The situation is similar at a negative crossing (the right of Fig. . The triple $(x,y,z)$ is called the ordered triple of colors at the crossing. Specifically, $x$ is the color of the source region, $y$ is the color of the under-arc from which the normal of the over-arc points, and $z$ is the color of the over-arc.