To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
Show all languages
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.

Tait conjectures

From Wikipedia, the free encyclopedia

The Tait conjectures are three conjectures made by 19th-century mathematician Peter Guthrie Tait in his study of knots.[1] The Tait conjectures involve concepts in knot theory such as alternating knots, chirality, and writhe. All of the Tait conjectures have been solved, the most recent being the Flyping conjecture.

YouTube Encyclopedic

  • 1/3
    17 952
    1 179
    1 707
  • ✪ The Millennium Prize Problems I
  • ✪ On integral aspects of the Tate conjecture - Alena Pirutka
  • ✪ Serre's Conjectures - Benedict Gross




A reduced diagram is one in which all the isthmi are removed.
A reduced diagram is one in which all the isthmi are removed.

Tait came up with his conjectures after his attempt to tabulate all knots in the late 19th century. As a founder of the field of knot theory, his work lacks a mathematically rigorous framework, and it is unclear whether he intended the conjectures to apply to all knots, or just to alternating knots. It turns out that most of them are only true for alternating knots.[2] In the Tait conjectures, a knot diagram is called "reduced" if all the "isthmi", or "nugatory crossings" have been removed.

Crossing number of alternating knots

Tait conjectured that in certain circumstances, crossing number was a knot invariant, specifically:

Any reduced diagram of an alternating link has the fewest possible crossings.

In other words, the crossing number of a reduced, alternating link is an invariant of the knot. This conjecture was proved by Louis Kauffman, Kunio Murasugi (村杉 邦男), and Morwen Thistlethwaite in 1987, using the Jones polynomial.[3] [4] [5] A geometric proof, not using knot polynomials, was given in 2017 by Joshua Greene.[6]

Writhe and chirality

A second conjecture of Tait:

An amphicheiral (or acheiral) alternating link has zero writhe.

This conjecture was also proved by Kauffman and Thistlethwaite.[3][7]


A flype move.
A flype move.

The Tait flyping conjecture can be stated:

Given any two reduced alternating diagrams and of an oriented, prime alternating link: may be transformed to by means of a sequence of certain simple moves called flypes.[8]

The Tait flyping conjecture was proved by Thistlethwaite and William Menasco in 1991.[9] The Tait flyping conjecture implies some more of Tait's conjectures:

Any two reduced diagrams of the same alternating knot have the same writhe.

This follows because flyping preserves writhe. This was proved earlier by Murasugi and Thistlethwaite.[10][7] It also follows from Greene's work.[6] For non-alternating knots this conjecture is not true; the Perko pair is a counterexample.[2] This result also implies the following conjecture:

Alternating amphicheiral knots have even crossing number.[2]

This follows because a knot's mirror image has opposite writhe. This conjecture is again only true for alternating knots: non-alternating amphichiral knot with crossing number 15 exist.[11]

See also


  1. ^ Lickorish, W. B. Raymond (1997), An introduction to knot theory, Graduate Texts in Mathematics, 175, Springer-Verlag, New York, p. 47, doi:10.1007/978-1-4612-0691-0, ISBN 978-0-387-98254-0, MR 1472978.
  2. ^ a b c Alexander Stoimenow, "Tait's conjectures and odd amphicheiral knots", Bull. Amer. Math. Soc. (N.S.) 45 (2008), no. 2, 285–291.
  3. ^ a b Kauffman, Louis (1987). "State models and the Jones polynomial". Topology. 26 (3): 395–407.
  4. ^ Murasugi, Kunio (1987). "Jones polynomials and classical conjectures in knot theory". Topology. 26 (2): 187–194.
  5. ^ Thistlethwaite, Morwen (1987). "A spanning tree expansion of the Jones polynomial". Topology. 26 (3): 297–309.
  6. ^ a b Greene, Joshua (2017). "Alternating links and definite surfaces". Duke Mathematical Journal. 166 (11): 2133–2151.
  7. ^ a b Thistlethwaite, Morwen (1988). "Kauffman's polynomial and alternating links". Topology. 27 (3): 311–318.
  8. ^ Weisstein, Eric W. "Tait's Knot Conjectures". MathWorld.
  9. ^ Menasco, William; Thistlethwaite, Morwen (1993). "The Classification of Alternating Links". Annals of Mathematics. 138 (1): 113–171.
  10. ^ Murasugi, Kunio (1987). "Jones polynomials and classical conjectures in knot theory. II". Mathematical Proceedings of the Cambridge Philosophical Society. 102 (2): 317–318.
  11. ^ Weisstein, Eric W. "Amphichiral Knot". MathWorld.
This page was last edited on 7 March 2019, at 04:39
Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.