Talk:Tait conjectures

Error

"A reduced alternating link with zero writhe is chiral.[2]"

Should be "amphichiral" — Preceding unsigned comment added by 75.25.17.33 (talk) 18:09, 15 April 2010‎

Actually, only the converse of the supposedly corrected version is true. The statement should read "A reduced alternating link with non-zero writhe is chiral." -- Ken Perko (lbrtpl@gmail.com) — Preceding unsigned comment added by 69.113.196.92 (talk) 13:52, 31 July 2014 (UTC)

An anonymous editor left essentially the same comment within the article today, which I am moving here: "The stated result is wrong. Acheiral implies zero writhe (for alternating knots; cf. Thistlethwaite's 15-crossing amphicheiral and Kidwell & Stoimenow's 16-crossing example with minimally projected writhes of 2, 0 and -2 [Mich.Math.J.51 (2003) 10]). Zero writhe does NOT imply acheiral." Additionally, I think there's a different problem with the article: much of it claims to be sourced to Kauffman's "Formal Knot Theory", but without page numbers, and doesn't seem to actually be mentioned in Kauffman's book. According to Google books, the book does not include the words "Tait", "writhe", "achiral", or "Thistlethwaite". Additionally while it contains many instances of the word "crossing" it does not seem to mention crossing numbers. A quick skim of the book doesn't find anything relevant. I think we need to find better sources. —David Eppstein (talk) 23:38, 11 September 2014 (UTC)

'Tis I (Ken Perko). Forget all that stuff about non-alternating amphicheirals (which belongs elsewhere) and just flip the second conjecture around to read something like "An amphicheiral (or acheiral) alternating link has zero writhe." Prior editing got this one backwards. I'm sure you can find sources in Lickorish's text for this well known result which goes back to Kauffman-Thistlethwaite-Murasugi's work with the Jones polynomial. — Preceding unsigned comment added by 69.113.194.237 (talk) 05:41, 12 September 2014 (UTC)

Ok, I'll do that. As for the source, for now I'll take out the problematic one and mark it as needing better. —David Eppstein (talk) 06:03, 12 September 2014 (UTC)

I think page 47 of W.B.R.Lickorish, "An Introduction to Knot Theory" (Springer, 1997) is what you're looking for. (Note that he refers to Tait's "conjectures" in quotes. He tells me he will insert a discussion of Little's theorem if there is ever a second edition.)--Ken Perko — Preceding unsigned comment added by 69.113.194.237 (talk) 20:35, 12 September 2014 (UTC)

Yes, that one looks much better. I've added it. —David Eppstein (talk) 21:14, 12 September 2014 (UTC)

The real problem with all this "Tait conjecture" stuff is that it ignores the fact that Tait (and Dehn and Heegaard) believed in Little's false "Theorem." on writhe, which most professionals have ignored, but which is explained by Epple at page 156 of "Die Entstehung der Knotentheorie." Perhaps when it's translated into English they'll wake up and smell the roses. (Ken Perko, lbrtpl@gmail,com)

Posibly dodgy paragraph

I've moved this unsourced material here from the Writhe and chirality section:

Tait never made the so-called "Tait conjectures" attributed to him long after his death and thus far proved only for alternating knots. He (along with Dehn and Heegaard) thought that Little had proven the invariance of writhe for minimal crossing diagrams of non-alternating knots as well, which seemed to be true for the knots in their tables. See M. Epple's "Die Entstehung der Knotentheorie" (1999) which reproduces in full Little's false proof in a 1900 paper placed in Trans. Roy. Soc. Edinburgh by Tait. The "Perko Pair" is the first counterexample for non-alternating knots and three more appear in Conway's 1970 extension of the Tait/Little tables to include all but 4 of the non-alternating 11-crossing knots. Another twenty-five counterexamples (named "Perko knots" in "LinKnot" (World Scientific, 2006)) have been identified by Jablan and Sazdanovic among the thousands of 12-crossing knots. In fact, the writhe of a reduced diagram is invariant for what seems to be the vast majority of non-alternating knots -- i.e., all those that are not Perko knots -- but nobody knows why. In this respect, Tait's "conjecture" on writhe remains unproved and remarkably un-understood.

The claim that Tait didn't make these conjectures is flat out contradicted by reference 1, and I'm not sure if any of it is directly relevant to this article. Someone who actually knows about the subject ought to have a look at it. 2.99.207.64 (talk) 20:54, 30 August 2016 (UTC)

DODGY? NONSENSE! Read Epple's account of Little's "Theorem" (pp. 154-156 of "Die Entstehung der Knotentheorie"). --Ken Perko, lbrtpl@gamil.com (someone who actually knows about the subject) — Preceding unsigned comment added by 24.184.200.119 (talk) 19:21, 31 December 2016 (UTC) Reference 1 makes clear that the conjectures attributed to Tait are at best implicit (and actually fall short in two respects: 1. most apply to adequate knots, not just alternating ones, and 2, The one on writhe of minimal crossing diagrams was adopted as true by Tait for all knots, which it is not: also it extends WAY beyond just alternating knots). Read some Epple you ignorant editors!