Talk:Tricolorability

message: It seems that the "Reidemeister III Move is 3-Colorable" figure should be replaced by a more appropriate one.  The 'before' and 'after' in the R-3 move should have the endpoints colored the same color (this is how one proves invariance under 3 colorability).  Simply showing two unrelated 3-colorings for the 'before' and 'after' does not seem helpful -- it may even confuse some. Adammanifold (talk) 21:29, 14 May 2009 (UTC)
replies: text: The above comment is correct (including the statement about confusion). In fact, it forces one the-same-color junction.  --90.180.192.165 (talk) 21:45, 26 October 2009 (UTC)
replies: text: Since others agree that this figure is wrong/misleading, I went ahead and created & uploaded a new one.  I also cleaned up this comments page so it is more readable overall. Adammanifold (talk) 02:18, 9 December 2009 (UTC)
thread_title:  Reidemeister III figure wrong? 

message: (This duplicates HIDDEN comment above)
Reidemaister Move II picture is not correct as well. Correct picture should have two parallel (uncrossed) strings at left ("before") picture, cf. Reidemeister_moves. --90.180.192.165 (talk) 21:45, 26 October 2009 (UTC)
replies: text: You're right -- the figure that was there was actually the Reidemeister move done twice!  I created & uploaded a correct figure. Adammanifold (talk) 02:38, 9 December 2009 (UTC)
thread_title:  R-2 Move 

message: >>Reidemeister Move I/II/III does not change tricolorability<< would be more precise than 
>>Reidemeister Move I/II/III is tricolorable<<. Moreover, a note might be added that >>Some other color combinations (e.g., all-the-same colors) should be considered for complete proof<< --90.180.192.165 (talk) 21:45, 26 October 2009 (UTC)
replies: text: I agree that this should be worded better.  I'll let you or someone else make these changes. Adammanifold (talk) 02:45, 9 December 2009 (UTC)
thread_title:  Another comment(s) 

message: Not only colorability, but hopefully also the number of possible colorings is a knot invariant.
I think I have seen it somewhere around here, and I will need it for links just today... --90.180.192.165 (talk) 21:45, 26 October 2009 (UTC)
replies: text: Tricolorability is a special case of Fox n-colorability (see Fox n-coloring).  This notion of counting the different number of possible colorings is covered there, so it probably doesn't need to be covered in this article too; at most it should be very briefly mentioned. Adammanifold (talk) 02:48, 9 December 2009 (UTC)
thread_title:  And a final comment 

message: The properties section contained the following sentence:
Any link with a separable component is also tricolorable.
I read this as saying that all separable links are tricolorable, which is not true (the crossingless diagram for the 2-component unlink cannot be colored using all three colors).  [Actually, it is true -- says I, Ken Perko, an expert on 3-colored covering spaces (see below)] I changed this to the following statement, because I can't think of what else the original author was trying to convey (the added words are in bold).
Any separable link with a tricolorable separable component is also tricolorable.
The above is true, but I feel that the first occurrence of the word separable might be superfluous.  If anyone else feels the same way, feel free to remove it.
Adammanifold (talk) 17:11, 28 June 2010 (UTC)
thread_title:  Properties section: separable component of a link 

message: The how to is an algorithm that is incomplete as it stands and there is doubt a greedy algorithm like this would always work without backtrack. Is there a citation or was this just made up? Dmcq (talk) 16:38, 17 July 2010 (UTC)
thread_title:  How to color a knot citation 

message: Substituted at 02:39, 5 May 2016 (UTC)
thread_title: Assessment comment

message: I saw some content which looks more like a comment than actually something that should appear in the article page, so I move them here:
*Actually, tricolorability is binary only for knots. For links there are other possibilities (which I call "partial tricolorability" -- where some components are left uncolored). See, for example, the Borromean Rings partial 3-coloring at the end of my 2014 paper "Remarks on the History of the Classification of Knots."  --Ken Perko (In fact, it's only binary for knots with fewer than three bridges, if one takes account of the number of different 3-colorings modulo mere renumbering of the colors.)
Z423x5c6 (talk) 13:35, 24 January 2022 (UTC)
thread_title:  Comments moved from the article page