Positive braids minimize ascending number

Lowell Davis; Jeffrey Meier

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arXiv · 2409.15490

Positive braids minimize ascending number

Lowell Davis·Jeffrey Meier

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Abstract

A well-known algorithm for unknotting knots involves traversing a knot diagram and changing each crossing that is first encountered from below. The minimal number of crossings changed in this way across all diagrams for a knot is called the ascending number of the knot. The ascending number is bounded below by the unknotting number. We show that for knots obtained as the closure of a positive braid, the ascending number equals the unknotting number. We also present data indicating that a similar result may hold for positive knots. We use this data to examine which low-crossing knots have the property that their ascending number is realized in a minimal crossing diagram, showing that there are at most 5 hyperbolic, alternating knots with at most 12 crossings with this property.

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BibTeX RIS

Lowell Davis, Jeffrey Meier. 2024-09-23. Positive braids minimize ascending number. https://arxiv.org/abs/2409.15490

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