Non-Equivalent knots which are indistinguishable by colourings

Question

Do there exist two non-equivalent knots which are indistinguishable by Fox $n$-colouring for every positive integer $n$?

That is, do there exist non-oriented knots $K_0$ and $K_1$ which are different (here I would like to exclude the case that one is the mirror image of the other) and for every $n\in\mathbb{Z}^+$, $\mathrm{col}_n(K_0)=\mathrm{col}_n(K_1)$, where $\mathrm{col}_n(K)$ denotes the number of distinct Fox $n$-colourings of a knot $K$?

How about the case of links?

Answer 1

I just realised that the knot $10_{124}$ and $10_{153}$ both have determinant $1$ and therefore, indistinguishable by Fox $n-$colouring for every positive integer $n$ since both cannot be coloured modulo $n$ for any $n$.

Answer 2

These are not prime knots, but the square knot and the granny knot I believe are indistinguishable by coloring. That is, $3_1\#\bar{3}_1$ and $3_1\#3_1$. You need peripheral subgroups (or at least that was the way it was done first). And they can be colored, if that helps your work.