Positive braids minimize ascending number
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866912047017492480 |
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| author | Davis, Lowell Meier, Jeffrey |
| author_facet | Davis, Lowell Meier, Jeffrey |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_15490 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Positive braids minimize ascending number Davis, Lowell Meier, Jeffrey Geometric Topology 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. |
| title | Positive braids minimize ascending number |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2409.15490 |