Let $D$ be a knot diagram and $Q$ a quandle. We use $c$ to denote a fixed coloring of $D$ with $Q$. If $D'$ is another knot diagram of the same knot, and $R_1$ is a sequence of Reidemeister moves connecting $D$ and $D'$. Obviously $R_1$ decides a unique $Q$-coloring of $D'$.
My question is, if $R_2$ is another sequence of Reidemeister moves connecting $D$ and $D'$, then $R_2$ will aslo bring a $Q$-coloring to $D'$. Are these two colorings equivalent?
Here we say two colorings of a knot diagram are equivalent if one can be transformed into the other one by some isotopy on the plane, i.e. no Reidemeister moves. For example let us consider the Fox 3-colorings on the standard diagram of trefoil knot, then there are five different colorings: three trivial colorings and two nontrivial colorings.