Slicing knots in definite 4-manifolds
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866915229951066112 |
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| author | Kjuchukova, Alexandra Miller, Allison N. Ray, Arunima Sakallı, Sümeyra |
| author_facet | Kjuchukova, Alexandra Miller, Allison N. Ray, Arunima Sakallı, Sümeyra |
| contents | We study the $\mathbb{CP}^2$-slicing number of knots, i.e. the smallest $m\geq 0$ such that a knot $K\subseteq S^3$ bounds a properly embedded, null-homologous disk in a punctured connected sum $(\#^m\mathbb{CP}^2)^{\times}$. We give a lower bound on the smooth $\mathbb{CP}^2$-slicing number of a knot in terms of its double branched cover, and we find knots with arbitrarily large but finite smooth $\mathbb{CP}^2$-slicing number. We also give an upper bound on the topological $\mathbb{CP}^2$-slicing number in terms of the Seifert form and find knots for which the smooth and topological $\mathbb{CP}^2$-slicing numbers are both finite, nonzero, and distinct. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_14596 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Slicing knots in definite 4-manifolds Kjuchukova, Alexandra Miller, Allison N. Ray, Arunima Sakallı, Sümeyra Geometric Topology 57K10, 57N35, 57N70, 57R40 We study the $\mathbb{CP}^2$-slicing number of knots, i.e. the smallest $m\geq 0$ such that a knot $K\subseteq S^3$ bounds a properly embedded, null-homologous disk in a punctured connected sum $(\#^m\mathbb{CP}^2)^{\times}$. We give a lower bound on the smooth $\mathbb{CP}^2$-slicing number of a knot in terms of its double branched cover, and we find knots with arbitrarily large but finite smooth $\mathbb{CP}^2$-slicing number. We also give an upper bound on the topological $\mathbb{CP}^2$-slicing number in terms of the Seifert form and find knots for which the smooth and topological $\mathbb{CP}^2$-slicing numbers are both finite, nonzero, and distinct. |
| title | Slicing knots in definite 4-manifolds |
| topic | Geometric Topology 57K10, 57N35, 57N70, 57R40 |
| url | https://arxiv.org/abs/2112.14596 |