Tait conjectures

Remarks

Part 1 was proved in 1987 by Kauffman, Murasugi and Thistlethwaite, independently, with the Jones polynomial. Part 2 was proved soon after, also with polynomial invariants. The key fact is that the span of the Jones polynomial is at most the number of crossings of any connected diagram, with equality for reduced alternating diagrams. Part 3, the flyping conjecture, was proved by Menasco and Thistlethwaite. Example: the standard diagram of the figure-eight knot is reduced and alternating, so its crossing number is $4$ and its writhe $0$ is the same in every reduced alternating diagram. The hypothesis that the diagram is reduced is needed: adding a nugatory kink keeps a diagram alternating but adds a crossing. Part 2 shows that an alternating knot with a reduced alternating diagram of nonzero writhe is not isotopic to its mirror image.

Going deeper

For a student

The theorems make the tabulation of alternating knots possible: crossing number is read off any reduced alternating diagram, and diagrams can be compared through flypes. The proof of part 1 is an early triumph of the Jones polynomial and of the Kauffman bracket state sum. For a reduced alternating diagram, the extreme states of the bracket each contribute a single monomial that cannot cancel. The writhe result gives a quick test for chirality of alternating knots. For non-alternating knots, finding the crossing number is much harder.