A note on surfaces in $\mathbb{CP}^2$ and $\mathbb{CP}^2\# \mathbb{CP}^2$
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866915229994057728 |
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| author | Marengon, Marco Miller, Allison N. Ray, Arunima Stipsicz, András I. |
| author_facet | Marengon, Marco Miller, Allison N. Ray, Arunima Stipsicz, András I. |
| contents | In this brief note, we investigate the $\mathbb{CP}^2$-genus of knots, i.e. the least genus of a smooth, compact, orientable surface in $\mathbb{CP}^2\setminus \mathring{B^4}$ bounded by a knot in $S^3$. We show that this quantity is unbounded, unlike its topological counterpart. We also investigate the $\mathbb{CP}^2$-genus of torus knots. We apply these results to improve the minimal genus bound for some homology classes in $\mathbb{CP}^2\# \mathbb{CP}^2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_12486 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A note on surfaces in $\mathbb{CP}^2$ and $\mathbb{CP}^2\# \mathbb{CP}^2$ Marengon, Marco Miller, Allison N. Ray, Arunima Stipsicz, András I. Geometric Topology 57R40, 57K10, 57N70 In this brief note, we investigate the $\mathbb{CP}^2$-genus of knots, i.e. the least genus of a smooth, compact, orientable surface in $\mathbb{CP}^2\setminus \mathring{B^4}$ bounded by a knot in $S^3$. We show that this quantity is unbounded, unlike its topological counterpart. We also investigate the $\mathbb{CP}^2$-genus of torus knots. We apply these results to improve the minimal genus bound for some homology classes in $\mathbb{CP}^2\# \mathbb{CP}^2$. |
| title | A note on surfaces in $\mathbb{CP}^2$ and $\mathbb{CP}^2\# \mathbb{CP}^2$ |
| topic | Geometric Topology 57R40, 57K10, 57N70 |
| url | https://arxiv.org/abs/2210.12486 |