A geometric proof of the flyping theorem

Abstract

In 1898, Tait asserted several properties of alternating knot diagrams. These assertions became known as Tait's conjectures and remained open until 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 (and independently Howie) answered a longstanding question of Fox by characterizing alternating links geometrically. Greene then used his characterization to give the first {\it geometric} proof of part of Tait's conjectures. We use Greene's characterization, Menasco's crossing ball structures, and a hierarchy of isotopy and {\it re-plumbing} moves to give the first entirely geometric proof of Menasco--Thistlethwaite's flyping theorem.