6-coloring of a knot

Question

According to this page this knot should be 6-colorable (question 6):

But I couldn't find an explicit coloring, which makes me think that the claim in the parantheses is not true. Can you find one?

In fact, if such a 6-coloring exists, then $$b+c=2a \mod6 \\ a+b=2c \mod 6 \\ c+a=2b \mod6.$$

Answer 1

If you are using Fox $n$-coloring, you should see that if a knot is $n$-colorable, then it is $kn$-colorable for all $k\in \mathbb{N}$. The knot in your picture is the trefoil, so it is 3-colorable, with $a=0, b=1,c=2$. To see it is 6-colorable just double all of these and you get a non-trivial 6-coloring.

For more information about Fox $n$-coloring, look at this paper by Jozef H. Przytycki. I believe that the "open" question mentioned in your link is actually solved now, but I may be mistaken.

Answer 2

Some approaches to knot colouring allow an extra colour to be added where a strand coloured A overlaps strand B with an open loop and then returns. The small section of B between the two parts of the loop coloured A can then be given a third colour C.

If this rather loose definition of colouring is used we can easily give the above knot six colours in a trivial sense by dragging two loops over from the outside to overlap other sections, then relabelling the overlapped parts inside and on one side of the overlappings until we get to six colours.

Answer 3

I don't buy Owad's answer. His 6-coloring isn't onto and is really just the same as the original 3-coloring; not trivial, but also not covering all six bases. I suspect the 6-colorability claim is just wrong; it surely is for the "dihedral" colorings that Fox was talking about, where even numbered colorings exist only for links (not knots). Perhaps there is some new palette of 6 colors that I'm not aware of. I have my own method of 6-coloring, which corresponds to the 4-fold "simple" covering spaces of knots. The colors are + or - red, blue green and it works whenever (and only when) the 3-colored branched covering space is not a Z2 homology sphere. The rules for +/- at crossings are easy and I've written it up on "Research Gate, Ken Perko."

Answer 4

And then there's "partial coloring" for links, where one or more componants gets left uncolored. I used a covering space corresponding to a partial 3-coloring of the Boromean rings to prove them unseparable (because it has less than the expected number of branches) at the end of my 2014 paper "Remarks on the History of the Classification of Knots." --Ken Perko, [email protected].