The equivariant genera of marked strongly invertible knots associated with $2$-bridge knots
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866911885385793536 |
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| author | Hirasawa, Mikami Hiura, Ryota Sakuma, Makoto |
| author_facet | Hirasawa, Mikami Hiura, Ryota Sakuma, Makoto |
| contents | A marked strongly invertible knot is a triple $(K,h,δ)$ of a knot $K$ in $S^3$, a strong inversion $h$ of $K$, and a subarc $δ\subset \operatorname{Fix}(h)\cong S^1$ bounded by $\operatorname{Fix}(h)\cap K\cong S^0$. An invariant Seifert surface for $(K,h,δ)$ is an $h$-invariant Seifert surface for $K$ that intersects $\operatorname{Fix}(h)$ in the arc $δ$. In this paper, we completely determine the equivariant genus (the minimum of the genera of invariant Seifert surfaces for $(K,h,δ)$) of every marked strongly invertible knot $(K,h,δ)$ with $K$ a $2$-bridge knot. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_06156 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The equivariant genera of marked strongly invertible knots associated with $2$-bridge knots Hirasawa, Mikami Hiura, Ryota Sakuma, Makoto Geometric Topology Primary 57K10, Secondary 57M60 A marked strongly invertible knot is a triple $(K,h,δ)$ of a knot $K$ in $S^3$, a strong inversion $h$ of $K$, and a subarc $δ\subset \operatorname{Fix}(h)\cong S^1$ bounded by $\operatorname{Fix}(h)\cap K\cong S^0$. An invariant Seifert surface for $(K,h,δ)$ is an $h$-invariant Seifert surface for $K$ that intersects $\operatorname{Fix}(h)$ in the arc $δ$. In this paper, we completely determine the equivariant genus (the minimum of the genera of invariant Seifert surfaces for $(K,h,δ)$) of every marked strongly invertible knot $(K,h,δ)$ with $K$ a $2$-bridge knot. |
| title | The equivariant genera of marked strongly invertible knots associated with $2$-bridge knots |
| topic | Geometric Topology Primary 57K10, Secondary 57M60 |
| url | https://arxiv.org/abs/2312.06156 |