Tait conjectures

This article is about knot theory. For the conjecture in graph theory, see Tait's conjecture.

The Tait conjectures are three conjectures made by 19th-century mathematician Peter Guthrie Tait in his study of knots.[1] 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

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 the conjectures apply to all knots, or just to alternating knots. Most of them are only true for alternating knots.[2] In the Tait conjectures, a knot diagram is reduced if all the isthmi (nugatory crossings) have been removed.

Crossing number of alternating knots

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 K. Murasugi in 1987, using the Jones polynomial.[1]

Writhe and chirality

A second conjecture of Tait:

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

This conjecture was also proven by Morwen Thistlethwaite.[1]

Tait never made the so-called "Tait conjectures" attributed to him long after his death and thus far proved only for alternating knots. He (along with Dehn and Heegaard) thought that Little had proven the invariance of writhe for minimal crossing diagrams of non-alternating knots as well, which seemed to be true for the knots in their tables. See M. Epple's "Die Entstehung der Knotentheorie" (1999) which reproduces in full Little's false proof in a 1900 paper placed in Trans. Roy. Soc. Edinburgh by Tait. The "Perko Pair" is the first counterexample for non-alternating knots and three more appear in Conway's 1970 extension of the Tait/Little tables to include all but 4 of the non-alternating 11-crossing knots. Another twenty-five counterexamples (named "Perko knots" in "LinKnot" (World Scientific, 2006)) have been identified by Jablan and Sazdanovic among the thousands of 12-crossing knots. In fact, the writhe of a reduced diagram is invariant for what seems to be the vast majority of non-alternating knots -- i.e., all those that are not Perko knots -- but nobody knows why. In this respect, Tait's "conjecture" on writhe remains unproved and remarkably un-understood.

Flyping

A flype move.

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.[3]

The Tait flyping conjecture was proven by Morwen Thistlethwaite and William Menasco in 1991.[3] 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.[2] The flyping conjecture also implies this conjecture:

Alternating amphicheiral knots have even crossing number.[2]

This follows because a knot's mirror image has opposite writhe.[2] This one is also only true for alternating knots, a non-alternating amphichiral knot with crossing number 15 was found, by Morwen Thistlethwaite.[4] Kidwell and Stoimenow found a 16-crossing amphicheiral knot with three different writhes: -2, 0 and 2.

See also

References

  1. 1 2 3 Lickorish, W. B. Raymond (1997), An introduction to knot theory, Graduate Texts in Mathematics 175, Springer-Verlag, New York, p. 47, doi:10.1007/978-1-4612-0691-0, ISBN 0-387-98254-X, MR 1472978.
  2. 1 2 3 4 A. Stoimenow, "Tait's conjectures and odd amphicheiral knots", 2007, arXiv: 0704.1941v1
  3. 1 2 Weisstein, Eric W., "Tait's Knot Conjectures", MathWorld.
  4. ↑ Weisstein, Eric W., "Amphichiral Knot", MathWorld.
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