Tait conjectures

Image:Isthmus.svg

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.

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.{{cite journal |last1=Kauffman |first1=Louis |authorlink=Louis Kauffman| date=1987 |title=State models and the Jones polynomial |journal=Topology |volume=26 |issue=3 |pages=395–407|doi=10.1016/0040-9383(87)90009-7 |doi-access=free }}

{{cite journal |last1=Murasugi |first1=Kunio |date=1987 |title=Jones polynomials and classical conjectures in knot theory |journal=Topology |volume=26 |issue=2 |pages=187–194|doi=10.1016/0040-9383(87)90058-9 |doi-access=free }}

{{cite journal |last1=Thistlethwaite |first1=Morwen |authorlink=Morwen Thistlethwaite| date=1987 |title=A spanning tree expansion of the Jones polynomial |journal=Topology |volume=26 |issue=3 |pages=297–309|doi=10.1016/0040-9383(87)90003-6 |doi-access=free }}

A geometric proof, not using knot polynomials, was given in 2017 by Joshua Greene.{{cite journal |last1=Greene |first1=Joshua |date=2017 |title=Alternating links and definite surfaces |journal=Duke Mathematical Journal |volume=166 |issue=11 |pages=2133–2151|bibcode=2015arXiv151106329G |arxiv=1511.06329 |doi=10.1215/00127094-2017-0004 |s2cid=59023367 }}

A second conjecture of Tait:

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

This conjecture was also proved by Kauffman and Thistlethwaite.

Image:Flype.svg move.]]

The Tait flyping conjecture can be stated:

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.{{MathWorld|TaitsKnotConjectures|Tait's Knot Conjectures}}

The Tait flyping conjecture was proved by Thistlethwaite and William Menasco in 1991.{{cite journal |last1=Menasco |first1=William |authorlink1=William Menasco| last2=Thistlethwaite |first2=Morwen | authorlink2=Morwen Thistlethwaite|date=1993 |title=The Classification of Alternating Links |journal=Annals of Mathematics |volume=138 |issue=1 |pages=113–171|jstor=2946636 |doi=10.2307/2946636 }}

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.{{cite journal |last1=Murasugi |first1=Kunio |date=1987 |title=Jones polynomials and classical conjectures in knot theory. II |journal=Mathematical Proceedings of the Cambridge Philosophical Society |volume=102 |issue=2 |pages=317–318|doi=10.1017/S0305004100067335 |bibcode=1987MPCPS.102..317M |s2cid=16269170 }}{{cite journal |last1=Thistlethwaite |first1=Morwen |date=1988 |title=Kauffman's polynomial and alternating links |journal=Topology |volume=27 |issue=3 |pages=311–318|doi=10.1016/0040-9383(88)90012-2 |doi-access=free }} 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:

Alternating amphicheiral knots have even crossing number.{{cite journal |first=Alexander |last=Stoimenow |title=Tait's conjectures and odd amphicheiral knots |journal=Bull. Amer. Math. Soc. |volume=45 |year=2008 |issue=2 |pages=285–291 |arxiv=0704.1941 |citeseerx=10.1.1.312.6024 |doi=10.1090/S0273-0979-08-01196-8 |s2cid=15299750 |url=https://www.ams.org/journals/bull/2008-45-02/S0273-0979-08-01196-8/}}

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.{{MathWorld|AmphichiralKnot|Amphichiral Knot}}

• Prime knot
• Tangle (knot theory)

{{reflist}}

{{Knot theory}}

Category:Conjectures that have been proved

Category:Knot theory