Tait conjectures are three or four statements as old as knot theory itself:
- Any connected, reduced (without isthmus) alternating diagram of a link has minimal number of crossings.
- Two reduced diagrams of an alternating knot have the same writhe. (or 2' for some: Acheiral alternating link has zero writhe, easy corollary of 2.)
- Two reduced, alternating diagrams of an oriented, prime link can be transformed into each other by a chain of flype moves.
- Knots with odd crossing number are not amphichiral.
Flype preserves writhe, so 3 implies 2 for alternating knots. Also 2 implies 4 for alternating knots because mirror of a knot has the same writhe with sign $\pm$ changed to $\mp$. I am aware of these but still looking for direct proofs.
Question 1. First conjecture was proven in [1] as Theorem 2.10. Wolfram Mathworld and other sources state that the same paper contains proof of second conjecture; I don't see it.
Statement of third one can be found in [2] on page 116 and then the proof follows.
Question 2. A non-alternating amphichiral knot with crossing number exactly 15 exists, so last conjecture needs to assume that the knot is alternating. Is there an explicit description of that knot available somewhere? What about higher crossing numbers? Do we know more counterexamples to the conjecture (without alternating assumption)?
[1] Louis Kauffman - State models and the Jones polynomial. https://mathscinet.ams.org/mathscinet-getitem?mr=899057
[2] Menasco, Thistlethwaite - The classification of alternating links https://mathscinet.ams.org/mathscinet-getitem?mr=1230928