|WikiProject Mathematics||(Rated Start-class, Low-priority)|
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)
Generalization to G-coloring
Because combinatorial aspects of the study of knot colorings are subsumed in the study of quandles, my feeling is that "Generalization to G-colorings" Section should focus on topological aspects, and definitions must be kept standard. This is a challenge, because I don't believe that there is a standard definition of a G-coloring in the literature. In particular, is a D2n-coloring a labeling of arcs by reflections (the way we have it now), by colors (red, blue, green,...), or by numbers from 1 to n (a Fox n-coloring)? If we can't resolve this, the section should be deleted. A Fox n-coloring is a pedagogical concept as opposed to the mathematical one (the representation ρ), and there is no corresponding pedagogical concept for a general group G.
If people want to keep the section, my proposed fix would be to differentiate between a D2n-coloring and a Fox n-coloring.
Dmoskovich (talk) 12:36, 5 January 2011 (UTC).
184.108.40.206 (talk) 19:55, 24 April 2017 (UTC)Ken Perko (LBRTPL@gmail.com): There is a great deal more to be said about Fox's dihedral n-colorings (which were actually discovered by Reidemeister), particularly the invariants derived by Kauffman from different diagrams of the same knot, such as the minimal number of different colors required, and the pallete of colors needed for various n. This article could be usefully expanded in that direction, based on a lot of current research, without going into the difficulties of comparison with various non-dihedral colorings. 24 April 2017.
Tricolor Diagram is Incorrect
The diagram illustrating "All Possible Tricolorings of the Trefoil Knot" is incorrect. The first row of three drawings showing the blue-red-green sequence is correct. In the second row, only the first drawing is correct for blue-green-red sequence. The second drawing in the second row is really the knot immediately above it (in the blue-red-green row) rotated 120 degrees. Same for the third drawing in the second row.