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.
Cite
@article{arxiv.2008.06490,
title = {A geometric proof of the flyping theorem},
author = {Thomas Kindred},
journal= {arXiv preprint arXiv:2008.06490},
year = {2024}
}
Comments
51 pages, 39 figures. There is an important change from version 2, where Theorem 4.9 was false. The overall proof strategy is the same, but implementing it requires an improved technique involving a hierarchy of moves. In the new version, the main argument concludes on page 33. Later pages are devoted to proofs of lemmas that appear earlier in the paper without proof. Comments are welcome