Knot diagram coloured with only one colour by any colouring

Question

Let $K$ be a knot diagram of a knot in $\mathbb{R}^3$. Suppose $K$ admits only trivial colourings by any quandle (a colouring is said to be trivial if only one colour is used to colour the diagram). Then is it true that $K$ is trivial; i.e. Unknotted

Answer 1

The fundamental quandle is a complete knot invariant. What this means, is that every nontrivial knot has a nontrivial coloring by its fundamental quandle.

So, to answer your question, yes. If a knot has only trivial colorings for every quandle, then it must be the trivial knot.

See Scott Carter's paper for more details.