The genus 1 bridge number of satellite knots
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866911061548990464 |
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| author | Taylor, Scott A. Tomova, Maggy |
| author_facet | Taylor, Scott A. Tomova, Maggy |
| contents | Let $T$ be a satellite knot, link, or spatial graph in a 3-manifold $M$ that is either $S^3$ or a lens space. Let $\mathfrak{b}_0$ and $\mathfrak{b}_1$ denote genus 0 and genus 1 bridge number, respectively. Suppose that $T$ has a companion knot $K$ (necessarily not the unknot) and wrapping number $ω$ with respect to $K$. When $K$ is not a torus knot, we show that $\mathfrak{b}_1(T)\geq ω\mathfrak{b}_1(K)$. There are previously known counter-examples if $K$ is a torus knot. Along the way, we generalize and give a new proof of Schubert's result that $\mathfrak{b}_0(T) \geq ω\mathfrak{b}_0(K)$. We also prove versions of the theorem applicable to when $T$ is a ``lensed satellite'' and when there is a torus separating components of $T$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_01834 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The genus 1 bridge number of satellite knots Taylor, Scott A. Tomova, Maggy Geometric Topology Let $T$ be a satellite knot, link, or spatial graph in a 3-manifold $M$ that is either $S^3$ or a lens space. Let $\mathfrak{b}_0$ and $\mathfrak{b}_1$ denote genus 0 and genus 1 bridge number, respectively. Suppose that $T$ has a companion knot $K$ (necessarily not the unknot) and wrapping number $ω$ with respect to $K$. When $K$ is not a torus knot, we show that $\mathfrak{b}_1(T)\geq ω\mathfrak{b}_1(K)$. There are previously known counter-examples if $K$ is a torus knot. Along the way, we generalize and give a new proof of Schubert's result that $\mathfrak{b}_0(T) \geq ω\mathfrak{b}_0(K)$. We also prove versions of the theorem applicable to when $T$ is a ``lensed satellite'' and when there is a torus separating components of $T$. |
| title | The genus 1 bridge number of satellite knots |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2401.01834 |