Flyping Conjecture

Also called the Tait flyping conjecture. Given two reduced alternating projections of the same knot, they are equivalent on the sphere iff they are related by a series of flypes. The conjecture was proved by Menasco and Thistlethwaite (1991, 1993) using properties of the Jones polynomial. It allows all possible reduced alternating projections of a given alternating knot to be drawn.

REFERENCES:

Adams, C. C. The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots. New York: W. H. Freeman, pp. 164-165, 1994.

Hoste, J.; Thistlethwaite, M.; and Weeks, J. "The First  Knots." Math. Intell. 20, 33-48, Fall 1998.

Menasco, W. and Thistlethwaite, M. "The Tait Flyping Conjecture." Bull. Amer. Math. Soc. 25, 403-412, 1991.

Menasco, W. and Thistlethwaite, M. "The Classification of Alternating Links." Ann. Math. 138, 113-171, 1993.

Stewart, I. The Problems of Mathematics, 2nd ed. Oxford, England: Oxford University Press, pp. 284-285, 1987.