Tait conjectures

Quality 0.50 · 1 view · Updated 2 months ago

title: "Tait conjectures" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["conjectures-that-have-been-proved", "knot-theory"] topic_path: "general/conjectures-that-have-been-proved" source: "https://en.wikipedia.org/wiki/Tait_conjectures" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

The Tait conjectures are three conjectures made by 19th-century mathematician Peter Guthrie Tait in his study of knots.{{citation | last = Lickorish | first = W. B. Raymond | authorlink=W. B. R. Lickorish | doi = 10.1007/978-1-4612-0691-0 | isbn = 978-0-387-98254-0 | mr = 1472978 | page = 47 | publisher = Springer-Verlag, New York | series = Graduate Texts in Mathematics | title = An introduction to knot theory | volume = 175 | year = 1997| s2cid = 122824389

Background

::figure[src="https://upload.wikimedia.org/wikipedia/commons/e/ec/Isthmus.svg" caption="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. 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: ::quote 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.

Writhe and chirality

A second conjecture of Tait: ::quote An amphicheiral (or acheiral) alternating link has zero writhe. ::

This conjecture was also proved by Kauffman and Thistlethwaite.

Flyping

::figure[src="https://upload.wikimedia.org/wikipedia/commons/d/da/Flype.svg" caption="A [[flype]] move."] ::

The Tait flyping conjecture can be stated: ::quote Given any two reduced alternating diagrams D_1 and D_2 of an oriented, prime alternating link: D_1 may be transformed to D_2 by means of a sequence of certain simple moves called flypes. ::

The Tait flyping conjecture was proved by Thistlethwaite and William Menasco in 1991. The Tait flyping conjecture implies some more of Tait's conjectures: ::quote 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. It also follows from Greene's work. For non-alternating knots this conjecture is not true; the Perko pair is a counterexample. This result also implies the following conjecture: ::quote Alternating amphicheiral knots have even crossing number. ::

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.

References

References

  1. (1987). "State models and the Jones polynomial". [[Topology (journal).
  2. (1987). "Jones polynomials and classical conjectures in knot theory". [[Topology (journal).
  3. (1987). "A spanning tree expansion of the Jones polynomial". Topology.
  4. {{MathWorld. TaitsKnotConjectures. Tait's Knot Conjectures
  5. (1993). "The Classification of Alternating Links". Annals of Mathematics.
  6. (1987). "Jones polynomials and classical conjectures in knot theory. II". [[Mathematical Proceedings of the Cambridge Philosophical Society]].
  7. (1988). "Kauffman's polynomial and alternating links". Topology.
  8. (2017). "Alternating links and definite surfaces". Duke Mathematical Journal.
  9. Stoimenow, Alexander. (2008). "Tait's conjectures and odd amphicheiral knots". Bull. Amer. Math. Soc..
  10. {{MathWorld. AmphichiralKnot. Amphichiral Knot

::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::

conjectures-that-have-been-provedknot-theory