On the tree-width of knot diagrams

Arnaud de Mesmay; Jessica Purcell; Saul Schleimer; Eric Sedgwick

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arXiv · 1809.02172

On the tree-width of knot diagrams

Arnaud de Mesmay·Jessica Purcell·Saul Schleimer·Eric Sedgwick

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Abstract

We show that a small tree-decomposition of a knot diagram induces a small sphere-decomposition of the corresponding knot. This, in turn, implies that the knot admits a small essential planar meridional surface or a small bridge sphere. We use this to give the first examples of knots where any diagram has high tree-width. This answers a question of Burton and of Makowsky and Mari\~no.

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Arnaud de Mesmay, Jessica Purcell, Saul Schleimer, Eric Sedgwick. 2018-09-06. On the tree-width of knot diagrams. https://arxiv.org/abs/1809.02172

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