Tait conjectures

[2] In the Tait conjectures, a knot diagram is called "reduced" if all the "isthmi", or "nugatory crossings" have been removed.

In other words, the crossing number of a reduced, alternating link is an invariant of the knot.

This conjecture was proved by Louis Kauffman, Kunio Murasugi (村杉 邦男), and Morwen Thistlethwaite in 1987, using the Jones polynomial.

[3][7] The Tait flyping conjecture can be stated: Given any two reduced alternating diagrams

[8] The Tait flyping conjecture was proved by Thistlethwaite and William Menasco in 1991.

[2] This result also implies the following conjecture: Alternating amphicheiral knots have even crossing number.