Geometry and Topology Seminar - Thomas Kindred - Flyping, replumbing, and symmetries of alternating links
Title: Flyping, replumbing, and symmetries of alternating links
Abstract: 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 gave the first geometric characterizations of alternating links, and Greene used his characterization to give 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 and sketch possible approaches to related facts which remain unproven by purely geometric means. I will also use spatial graphs, branched surfaces, and replumbing moves to describe what the flyping theorem implies about symmetries of alternating links.
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