Prove that the knots 4_1 and 5_2 are not equivalent by showing one is p-colorable and the other is not for some prime p.

Question

I've been learning about knot theory and I am a little confused how to prove the above statement. I know that the 4_1 knot (or the figure-eight knot) is not 3-colorable. However, it is 5-colorable. So when it says to prove it's p-colorable for some prime p, would it be considered p-colorable because it is 5-colorable? How would you prove this type of proposition?

Answer 1

Find a prime $p$ such that one of them is $p$-colorable and the other one not. In other words: For both knots, $p$ shall be the same.