Coloring of a knot diagram by a trivial quandle

Question

Suppose a knot diagram is colored by a trivial quandle $T_n$ of order $n$ (The trivial quandle $T_n$ is a finite set of order n with the binary operation $x*y=x$ for all $x,y \in T_n$ ). My question is: Is it necessary that $n=1$ in which case we say that the coloring is trivial? Or we can find a non-trivial coloring of a knot diagram by a trivial quandle?

Answer 1

I will assume you mean to say that the coloring uses every element of $T_n$. Colorings don't need to be surjective in general.

For links, no; for knots, yes.

For a link: consider the Hopf link. It's fundamental quandle is $T_2$.

For a knot: choose a diagram. Follow the knot around, labeling the strands $A_1,A_2,...,A_k$ (so $A_i$ and $A_{i+1}$ are opposite each other as understrands at a crossing). Let $a_1,\dots,a_n\in T_n$ be the corresponding colors. The relationship for a knot quandle is $a_i*a_j=a_{i+1}$ at a crossing with $A_j$ the overstrand and $A_i,A_{i+1}$ the understrands. The quandle is trivial so $a_i=a_{i+1}$. Hence, knots can only be colored surjectively by $T_1$.