Colorings of Knot Diagrams by Quandles

Let $X$ be a fixed quandle. Let $D$ be a given oriented classical knot or link diagram, and let ${\mathcal R}$ be the set of (over-)arcs. The normals (normal vectors) are given in such a way that the ordered pair (tangent, normal) agrees with the orientation of the plane (the ordered pair of the standard $x$- and $y$-axes), see Fig. .

Figure: Quandle relation at a crossing

A (quandle) coloring ${\mathcal C}$ is a map ${\mathcal C} : {\mathcal R} \rightarrow X$ such that at every crossing, the relation depicted in Fig.  holds. If the normal to the over-arc $\beta$ points from the arc $\alpha$ to $\gamma$, then it is required that ${\mathcal C}(\alpha)*{\mathcal C}(\beta)={\mathcal C}(\gamma)$. The ordered pair $(a,b)$ is called the source colors or ordered pair of colors at the crossing.

Let ${\rm Col}_X(D)$ denote the set of colorings of a knot diagram $D$ of a knot $K$ by a quandle $X$. It is known [FR92], and can be checked on diagrams, that for any coloring ${\mathcal C}$ of a diagram $D$ of a knot $K$ by a quandle $X$, there is a unique coloring ${\mathcal C}'$ of a diagram $D'$ of the same knot $K$ that is obtained from $D$ by a single Reidemeister move of any type (I, II, or III), such that the colorings coincide outside of a small portion of the diagrams where the move is performed. Thus there is a one-to-one correspondence between the sets of colorings of two diagrams of the same knot. In particular, the cardinality $\vert{\rm Col}_X(D)\vert$ is a knot invariant. Note that any knot diagram is colored by a single element of a given quandle. Such a monochromatic coloring is called trivial. Hence for any knot $K$ and any quandle $X$, $\vert{\rm Col}_X(D)\vert$ is at least the cardinality $\vert X\vert$ of the quandle.

In fact, the Reidemeister moves I, II, III correspond to the quandle axiom I, II, III, respectively. In particular, a type III move represents the self-distributivity as depicted in Fig. .

Figure: Type III move and self-distributivity

A coloring is defined for virtual knot diagrams in a completely similar manner, and the sets of colorings of two virtual diagrams of the same virtual knot are in one-to-one correspondence, as checked by Reidemeister moves for virtual knots.

For a coloring ${\mathcal C}$ of a knot diagram $D$, there is a coloring of regions ([FRS95,FRS*]) that extend ${\mathcal C}$ as depicted in Fig. . The set ${\mathcal Rg}$ of regions is defined to be the set of connected components of the complement of (underlying) projection of a knot diagram in $\mathbb{R}^2$ (or $\mathbb{S}^2$). A coloring of regions is an assignment of quandle elements to regions, together with a given coloring ${\mathcal C}$ of arcs, $\tilde{\mathcal C} : {\mathcal R} \cup {\mathcal Rg} \rightarrow X$, $\tilde{\mathcal C} \vert _{\mathcal R} = {\mathcal C}$, 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 $x * {\mathcal C}(\alpha)$. The set of colorings $\tilde{\mathcal C} : {\mathcal R} \cup {\mathcal Rg} \rightarrow X$ of arcs and regions is denoted by ${\rm Colr}_X(D)$.

Figure: Quandle colorings of regions

In Fig. , it is seen that a coloring of regions (region colors) are well-defined near a crossing. In the left of the figure, a positive crossing is depicted. Near a crossing, there are four regions, one of which is a unique region from which all the normal vectors of the arcs point to other regions. Such a region is called the source region at a crossing. 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 the self-distributivity of a quandle. The situation is similar at a negative crossing (the right of Fig. , the colors of the regions unspecified are for an exercise). The triple of elements $(x,y,z)$ of $X$ is called the source colors or 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 (the pair $(y,z)$ is the source colors of arcs).

For any coloring ${\mathcal C}$ of a knot diagram $D$ on $\mathbb{R}^2$ (or $\mathbb{S}^2$) by a quandle $X$, and any specific choice of a color $x_0 \in X$ for a region $R_0 \in {\mathcal Rg}$, there is a unique region coloring $\tilde{\mathcal C} : {\mathcal R} \cup {\mathcal Rg} \rightarrow X$ that extends ${\mathcal C}$, $\tilde{\mathcal C} \vert _{\mathcal R} = {\mathcal C}$, such that $\tilde{\mathcal C} ( R_0)=x_0$. This is seen as follows. Let $R \in {\mathcal Rg}$ and take a path $\gamma$ from $R_0$ to $R$ that avoids crossings of $D$ and intersects transversely in finite points with $D$. Along $\gamma$ the element $\tilde{\mathcal C} (R)$ is determined by the coloring rule of regions from the colors of arcs. Take two paths, $\gamma_1$ and $\gamma_2$, from $R_0$ to $R$, that miss crossing points. There is a homotopy from $\gamma_1$ to $\gamma_2$, and it is assumed without loss of generality that during the homotopy the paths experience, themselves and in relation the the knot projection, a sequence of Reidemeister moves. In particular, when a path goes through a crossing of the knot projection, it corresponds to a type III move, and well-definedness under such a move is checked in Fig. . Therefore the color of regions are uniquely determined by a color of a single region. Note that we used the fact that the plane is simply connected, i.e., any two paths are homotopic. In general, we cannot define region colors for knots on a surface, or virtual knots.