Hard Diagrams of the Unknot

Benjamin A. Burton; Hsien-Chih Chang; Maarten Löffler; Arnaud de Mesmay; Clément Maria; Saul Schleimer; Eric Sedgwick; Jonathan Spreer

arXiv · 2104.14076

Hard Diagrams of the Unknot

Benjamin A. Burton·Hsien-Chih Chang·Maarten Löffler·Arnaud de Mesmay·Clément Maria·Saul Schleimer·Eric Sedgwick·Jonathan Spreer

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Abstract

We present three "hard" diagrams of the unknot. They require (at least) three extra crossings before they can be simplified to the trivial unknot diagram via Reidemeister moves in $\mathbb{S}^2$. Both examples are constructed by applying previously proposed methods. The proof of their hardness uses significant computational resources. We also determine that no small "standard" example of a hard unknot diagram requires more than one extra crossing for Reidemeister moves in $\mathbb{S}^2$.

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Benjamin A. Burton, Hsien-Chih Chang, Maarten Löffler, Arnaud de Mesmay, Clément Maria, Saul Schleimer, Eric Sedgwick, Jonathan Spreer. 2021-04-29. Hard Diagrams of the Unknot. https://arxiv.org/abs/2104.14076

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