Positive Knots and Ribbon Concordance
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866915232796901376 |
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| author | Boninger, Joe |
| author_facet | Boninger, Joe |
| contents | Ribbon concordances between knots generalize the notion of ribbon knots. Agol, building on work of Gordon, proved ribbon concordance gives a partial order on knots in $S^3$. In previous work, the author and Greene conjectured that positive knots are minimal in this ordering. In this note we prove this conjecture for a large class of positive knots, and show that a positive knot cannot be expressed as a non-trivial band sum -- both results extend earlier theorems of Greene and the author for special alternating knots. In a related direction, we prove that if positive knots $K$ and $K'$ are concordant and $|σ(K)| \geq 2g(K) - 2$, then $K$ and $K'$ have isomorphic rational Alexander modules. This strengthens a result of Stoimenow, and gives evidence toward a conjecture that any concordance class contains at most one positive knot. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_08103 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Positive Knots and Ribbon Concordance Boninger, Joe Geometric Topology Ribbon concordances between knots generalize the notion of ribbon knots. Agol, building on work of Gordon, proved ribbon concordance gives a partial order on knots in $S^3$. In previous work, the author and Greene conjectured that positive knots are minimal in this ordering. In this note we prove this conjecture for a large class of positive knots, and show that a positive knot cannot be expressed as a non-trivial band sum -- both results extend earlier theorems of Greene and the author for special alternating knots. In a related direction, we prove that if positive knots $K$ and $K'$ are concordant and $|σ(K)| \geq 2g(K) - 2$, then $K$ and $K'$ have isomorphic rational Alexander modules. This strengthens a result of Stoimenow, and gives evidence toward a conjecture that any concordance class contains at most one positive knot. |
| title | Positive Knots and Ribbon Concordance |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2405.08103 |