Talk:Fox n-coloring
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[edit] Mistake in definition of n-coloring
There's a mistake in the definition of n-coloring. For example, if you take the nontrivial coloring of a knot, take two strands of different color that you can do a Reidemeister two move on, then after that move, you get a diagram that has a natural coloring induced by any coloring before the move, but now you can use the extra colors to get many more colorings by coloring the extra strand that arose from the move. So the property of having the same number of colorings would not be preserved under Reidemeister moves, which I believe is supposed to be a fundamental property that the definition should imply.
I believe there's a linear congruence relation (if a color is an element of integers mod n) that must be satisfied by the three strands at each crossing. Reidemeister one and two moves aren't a real problem for this, but the right rule should work for all three moves. I don't have time to check how this should work out, unfortunately. My Wikipedia time is out for the next couple days. --C S (Talk) 01:03, 20 May 2006 (UTC)
- Hm, took me forever, but I got to it...the definition is fixed and I will add a reference to a very nice intro to knot theory using coloring by Przytycki. It contains much more info for people interested in learning more about n-coloring and material to add to the article. --C S (Talk) 04:14, 28 September 2006 (UTC)
