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.
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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.
Menasco, along with Morwen Thistlethwaite proved the Tait flyping conjecture, which states that given any two reduced alternating diagrams D1,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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Morwen Thistlethwaite, Louis Kauffman and K. Murasugi proved the first two Tait conjectures in 1987 and Morwen Thistlethwaite and William Menasco proved the Tait flyping conjecture in 1991.