The Tait conjectures are three conjectures made by 19th-century mathematician

Peter Guthrie Tait Peter Guthrie Tait FRSE (28 April 1831 – 4 July 1901) was a Scottish mathematical physicist and early pioneer in thermodynamics. He is best known for the mathematical physics textbook ''Treatise on Natural Philosophy'', which he co-wrote wi ...

in his study of knots.. The Tait conjectures involve concepts in

knot theory In the mathematical field of topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot ...

such as

alternating knot In knot theory, a knot or link diagram is alternating if the crossings alternate under, over, under, over, as one travels along each component of the link. A link is alternating if it has an alternating diagram. Many of the knots with crossing ...

chirality Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable from i ...

, and

writhe In knot theory, there are several competing notions of the quantity writhe, or \operatorname. In one sense, it is purely a property of an oriented link diagram and assumes integer values. In another sense, it is a quantity that describes the amoun ...

. All of the Tait conjectures have been solved, the most recent being the Flyping conjecture.

Background

Tait came up with his conjectures after his attempt to

tabulate Tabulata, commonly known as tabulate corals, are an order of extinct forms of coral. They are almost always colonial, forming colonies of individual hexagonal cells known as corallites defined by a skeleton of calcite, similar in appearance to ...

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

s. 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 In the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some ...

, specifically:

Any reduced
diagram A diagram is a symbolic representation of information using visualization techniques. Diagrams have been used since prehistoric times on walls of caves, but became more prevalent during the Enlightenment. Sometimes, the technique uses a three ...
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 Louis Hirsch Kauffman (born February 3, 1945) is an American mathematician, topologist, and professor of mathematics in the Department of Mathematics, Statistics, and Computer science at the University of Illinois at Chicago. He is known for t ...

Kunio Murasugi Kunio (written: 邦夫, 邦男, 邦雄, 邦生, 國男, 國士, 国男, 国夫, 州男 or 久仁生) is a masculine Japanese given name. Notable people with the name include: *, Japanese businessman *, Japanese businessman *, Japanese judge *, Jap ...

(村杉邦男), and

Morwen Thistlethwaite Morwen Bernard Thistlethwaite is a knot theorist and professor of mathematics for the University of Tennessee in Knoxville. He has made important contributions to both knot theory and Rubik's Cube group theory. Biography Morwen Thistlethwait ...

in 1987, using the

Jones polynomial In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polyno ...

. A geometric proof, not using knot polynomials, was given in 2017 by Joshua Greene.

Writhe and chirality

A second conjecture of Tait:

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

This conjecture was also proved by

Kauffman Kaufmann is a surname with many variants such as Kauffmann, Kaufman, and Kauffman. In German, the name means ''merchant''. It is the cognate of the English '' Chapman'' (which had a similar meaning in the Middle Ages, though it disappeared from ...

and Thistlethwaite.

Flyping

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 ''
flype In the mathematical theory of knots, a flype is a kind of manipulation of knot and link diagrams used in the Tait flyping conjecture. It consists of twisting a part of a knot, a tangle T, by 180 degrees. Flype comes from a Scots word meaning '' ...
s''.

The Tait flyping conjecture was proved by Thistlethwaite and

William Menasco William W. Menasco is a topologist and a professor at the University at Buffalo. He is best known for his work in knot theory. Biography Menasco received his B.A. from the University of California, Los Angeles in 1975, and his Ph.D. from the Unive ...

in 1991. The Tait flyping conjecture implies some more of Tait's conjectures:

Any two reduced diagrams of the same alternating
knot A knot is an intentional complication in Rope, cordage which may be practical or decorative, or both. Practical knots are classified by function, including List of hitch knots, hitches, List of bend knots, bends, List of loop knots, loop knots, ...
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 In the mathematical theory of knots, the Perko pair, named after Kenneth Perko, is a pair of entries in classical knot tables that actually represent the same knot. In Dale Rolfsen's knot table, this supposed pair of distinct knots is labeled 1 ...

is a counterexample. This result also implies the following conjecture:

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

{{Knot theory Conjectures that have been proved Knot theory

Background

Crossing number of alternating knots

Writhe and chirality

Flyping

See also

References