Tricolorability of knots: Are there knots of arbitrary large k-colorability?

Question

The trefoil knot is tricolorable: In a projection, with at least two colors used to color its "strands," at every (degree-$3$) crossing, either all three colors come together, or all three strands have the same color. The figure-$8$ knot is not tricolorable: It needs $4$ colors.

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       ^{Image from Wikipedia article.}

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Q. Are there knots that require $k$ colors, for arbitrarily large $k$?

I assume the answer is Yes, in which case examples would be appreciated.

Answer 1

According to Breiland, Oesper, and Taalman, in this paper, Theorem 1 says:

Suppose $T_{m,n}$ is a torus knot and $p$ is prime.

1. If $m$ and $n$ are both odd, then $T_{m,n}$ is not $p$-colorable.
2. If $m$ is odd and $n$ is even, then $T_{m,n}$ is $p$-colorable if and only if $p|m$

This gives us that for any prime $m$, $T_{m,2}$ is $m$-colorable. I am sure that there are other good examples, but here is a positive answer to your question.

Answer 2

Just take the torus knot T_p,1 for arbitratily large p. It's p-colorable. --Ken Perko