A geometric proof of the flyping theorem

TL;DR

This paper provides the first purely geometric proof of the flyping theorem in knot theory, utilizing Greene's geometric characterization of alternating links and Menasco's crossing ball structures.

Contribution

It introduces a novel geometric proof of the flyping theorem, combining Greene's characterization with Menasco's structures and re-plumbing moves.

Findings

First geometric proof of the flyping theorem

Utilizes Greene's geometric characterization of alternating links

Integrates crossing ball structures and re-plumbing moves

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.