Tait conjectures

The Tait conjectures are conjectures made by Peter Guthrie Tait in his study of knots. 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 Tait flyping conjecture proven in 1991 by Morwen Thistlethwaite and William Menasco.

Background

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 the conjectures apply to all knots, or just to alternating knots. Most of them are only true for alternating knots. In the Tait conjectures, a knot diagram is reduced if all the isthmus have been removed.

The Tait conjectures

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 an reduced, alternating link is an invariant of the knot. This conjecture was proven by Morwen Thistlethwaite, Louis Kauffman and K. Murasugi in 1987, using the Jones polynomial. Another one of his conjectures:

A reduced alternating link with zero writhe implies that the link is chiral.Louis Kauffman, "Formal knot theory", 2006, ISBN 0-486-45052-X 221-227]

This conjecture was also proven by Morwen Thistlethwaite.

The Tait flyping conjecture

The Tait flyping conjecture can be stated:

Given any two reduced alternating diagrams D₁ and D₂ of an oriented, prime alternating link: D₁ may be transformed to D₂ by means of a sequence of certain simple moves called "flypes". Also known as the Tait flyping conjecture.Weisstein, Eric W. "Tait's Knot Conjectures." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/TaitsKnotConjectures.html]

The Tait flyping conjecture was proven by Morwen Thistlethwaite and William Menasco in 1991. 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 proven earlier by Morwen Thistlethwaite, Louis Kauffman and K. Murasugi in 1987. For non-alternating knots this conjecture is not true, assuming so lead to the duplication of the Perko pair, because it has two reduced projections with different writhe. The flyping conjecture also implies this conjecture:

Alternating Amphichiral knots have even crossing number. A. Stoimenow, "Tait's conjectures and odd amphicheiral knots", 2007, [http://arxiv.org/abs/0704.1941 arXiv: 0704.1941v1] ]

This follows because a knot's mirror image has opposite writhe. This one is also only true for alternating knots, a non-alternating amphichiral knot with crossing number 15 was found, by Morwen Thistlethwaite. [Weisstein, Eric W. "Amphichiral Knot." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/AmphichiralKnot.html]

References

ee also

*Knot theory
*Tangle (knot theory)
*Knot (mathematics)
*Prime knot