example of knot diagram colored by dihedral quandle of non-orime order, if any

Question

Is there a known example of knot colored by a dihedral quandle of non-prime order, for example the diherdral qunadle of order 4, 6 or 12.

Answer 1

Let $D(n)$ denote the dihedral quandle of order $n$

Then for any $m$ and $n$, the quandle $D(mn)$ contains copies of both $D(m)$ and $D(n)$. (Indeed, $D(mn)$ is the union of $m$ copies of $D(n)$, as well as $n$ copies of $D(m)$.) It follows that any coloring of a knot by either $D(m)$ or $D(n)$ also gives a coloring of the knot by $D(mn)$.

By the way, note that dihedral quandles of even order are disconnected. In particular, every coloring of a knot by a dihedral quandle of order $2n$ is really a coloring by one of its two dihedral subquandles of order $n$.

Of course, this doesn't answer the question of whether there are any onto colorings of knots by dihedral quandles of composite order. For example, is it possible to color a knot with $D(9)$ so that all nine colors are used? One can also ask whether there are any essential colorings of knots by $D(9)$, i.e. colorings in which the colors used do not all lie in some proper subquandle.

Answer 2

the torus knot (2,9) is equivalent to 9_1 with 9 crossings and it is colored by the dihedral quandle R_9 such that all the nine colors used.