Bounding the ribbon numbers of knots and links
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866914919138459648 |
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| author | Friedl, Stefan Misev, Filip Zupan, Alexander |
| author_facet | Friedl, Stefan Misev, Filip Zupan, Alexander |
| contents | The ribbon number $r(K)$ of a ribbon knot $K \subset S^3$ is the minimal number of ribbon intersections contained in any ribbon disk bounded by $K$. We find new lower bounds for $r(K)$ using $\det(K)$ and $Δ_K(t)$, and we prove that the set $\mathfrak{R}_r~=~\{Δ_K(t)~:~r(K)~\leq~r\}$ is finite and computable. We determine $\mathfrak{R}_2$ and $\mathfrak{R}_3$, applying our results to compute the ribbon numbers for all ribbon knots with 11 or fewer crossings, with three exceptions. Finally, we find lower bounds for ribbon numbers of links derived from their Jones polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_11618 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bounding the ribbon numbers of knots and links Friedl, Stefan Misev, Filip Zupan, Alexander Geometric Topology 57K10 The ribbon number $r(K)$ of a ribbon knot $K \subset S^3$ is the minimal number of ribbon intersections contained in any ribbon disk bounded by $K$. We find new lower bounds for $r(K)$ using $\det(K)$ and $Δ_K(t)$, and we prove that the set $\mathfrak{R}_r~=~\{Δ_K(t)~:~r(K)~\leq~r\}$ is finite and computable. We determine $\mathfrak{R}_2$ and $\mathfrak{R}_3$, applying our results to compute the ribbon numbers for all ribbon knots with 11 or fewer crossings, with three exceptions. Finally, we find lower bounds for ribbon numbers of links derived from their Jones polynomials. |
| title | Bounding the ribbon numbers of knots and links |
| topic | Geometric Topology 57K10 |
| url | https://arxiv.org/abs/2408.11618 |