Quandles and knot-diagram colorings

A quandle is a set $Q$ with a binary operation $\ast :Q\times Q\to Q$ such that:

1. $x\ast x=x$ for every $x\in Q$;
2. for every $y\in Q$, the right translation $R_y:Q\to Q$, $R_y(x)=x\ast y$, is a bijection;
3. $(x\ast y)\ast z=(x\ast z)\ast (y\ast z)$ for all $x,y,z\in Q$.

Write for the unique element satisfying . Let $D$ be a regular oriented link diagram. An arc of $D$ is a connected strand segment cut at each crossing where that strand passes under; an overstrand continues through its crossing. Orient the projection plane. Along an oriented overarc with tangent $t$, let $n$ be the unit normal in the plane for which $(t,n)$ is positively oriented. At each crossing, call the underarc from which $n$ points the source underarc and the underarc toward which $n$ points the target underarc. A quandle coloring of $D$ by $Q$ is an assignment $c$ of an element of $Q$ to every arc such that, at each crossing, if the source, over, and target colors are $a,b,d$, then $d=a\ast b$. Equivalently, when the source and target are named in the opposite order, . Write ${\mathrm{Col}}_Q(D)$ for the set of all such assignments.

The local convention is shown below. The overarc is oriented from southwest to northeast, so its positive normal points northwest; the underarc is oriented from southeast to northwest. Thus $x$ is the source color, $y$ is the overarc color, and $z=x\ast y$ is the target color. The gap marks the underpassing strand.

Regular oriented link diagrams.

Manturov, Knot Theory, Chapter 5.

Carter, A Survey of Quandle Ideas.

Przytycki, 3-coloring and other elementary invariants of knots.