Geometry and Topology Seminar
In 1898, P.G. Tait asserted several properties of alternating link diagrams, which remained unproven until the discovery of the Jones polynomial in 1985. By 1993, the Jones polynomial had led to proofs of all of Tait's conjectures, but the geometric content of these new results remained mysterious.
In 2017, Howie and Greene independently gave the first geometric characterizations of alternating links; as a corollary, Greene obtained the first purely geometric proof of part of Tait's conjectures. Recently, I used these characterizations and "replumbing" moves, among other techniques, to give the first entirely geometric proof of Tait's flyping conjecture, first proven in 1993 by Menasco and Thistlethwaite.
I will describe these recent developments, focusing in particular on the fundamentals of plumbing (also called Murasugi sum), and definite surfaces (which characterize alternating links a la Greene). As an aside, I will also sketch a (partly new, simplified) proof of the classical result of Murasugi and Crowell that the genus of an alternating knot equals half the degree of its Alexander polynomial. The talk will be broadly accessible. Expect lots of pictures!