Colorings of knot diagrams

Let $X$ be a fixed quandle. Let $K$ be a given oriented classical knot or link diagram, and let ${\cal R}$ 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 ${\cal C}$ is a map ${\cal C} : {\cal R} \rightarrow X$ 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 ${\cal C}(\alpha)*{\cal C}(\beta)={\cal C}(\gamma)$. The pair $(a,b)$ is called the ordered pair of colors at the crossing. Let ${\rm Col}_X(K)$ denote the set of colorings of a knot diagram $K$ by a quandle $X$. It is known [FeRou92] that there is a unique coloring after any Reidemeister move given a coloring before the move, and therefore, the cardinality $\vert{\rm Col}_X(K)\vert$ 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 $X$, $\vert{\rm Col}_X(K)\vert$ is at least the number of quandle, $\vert X\vert$.

When we use Alexander quandles to color diagrams, Inoue's Theorem [Ino01] is helpful. There is a knot invariant called Alexander polynomial, and an implication of Inoue's Theorem is that a knot is colored only trivially by an Alexander quandle $\mathbb{Z}_p[t, t^{-1}]/g(t)$ if its Alexander polynomial is coprime mod $p$ to $g(t)$.