| Abstract: In 1898, Tait asserted several properties of alternating knot diagrams. These assertions came to be known as Tait's conjectures and remained open through the discovery of the Jones polynomial in 1985. The new polynomial invariants soon led to proofs of all of Tait's conjectures, culminating in 1993 with Menasco-Thistlethwaite's proof of Tait's flyping conjecture. In 2017, Greene gave the first geometric proof of part of Tait's conjectures, while also answering a longstanding question of Fox by characterizing alternating links geometrically; Howie independently answered Fox's question with a related characterization. I will use these new characterizations, among other techniques, to give the first entirely geometric proof of Menasco-Thistlethwaite's Flyping Theorem. |