Tait conjectures
The Tait conjectures are three conjectures made by 19th-century mathematician 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.
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 isthmi (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 proven by Morwen Thistlethwaite, Louis Kauffman and Kunio Murasugi (村杉 邦男) in 1987, using the Jones polynomial.
A second conjecture of Tait:
An amphicheiral (or acheiral) alternating link has zero writhe.
This conjecture was also proven by Morwen Thistlethwaite.
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.
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