Branched covers of twist-roll spun knots and turned twisted tori
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866908292195811328 |
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| author | Hughes, Mark Kim, Seungwon Miller, Maggie |
| author_facet | Hughes, Mark Kim, Seungwon Miller, Maggie |
| contents | We prove that the double branched cover of a twist-roll spun knot in $S^4$ is smoothly preserved when four twists are added, and that the double branched cover of a twist-roll spun knot connected sum with a trivial projective plane is preserved after two twists are added. As a consequence, we conclude that the members of a family of homotopy $\mathbb{CP}^2$s recently constructed by Miyazawa are each diffeomorphic to $\mathbb{CP}^2$. We also apply our techniques to show that the double branched covers of odd-twisted turned tori are all diffeomorphic to $S^2 \times S^2$, and show that a family of homotopy 4-spheres constructed by Juhász and Powell are all diffeomorphic to $S^4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_11706 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Branched covers of twist-roll spun knots and turned twisted tori Hughes, Mark Kim, Seungwon Miller, Maggie Geometric Topology 57K45, 57K40 We prove that the double branched cover of a twist-roll spun knot in $S^4$ is smoothly preserved when four twists are added, and that the double branched cover of a twist-roll spun knot connected sum with a trivial projective plane is preserved after two twists are added. As a consequence, we conclude that the members of a family of homotopy $\mathbb{CP}^2$s recently constructed by Miyazawa are each diffeomorphic to $\mathbb{CP}^2$. We also apply our techniques to show that the double branched covers of odd-twisted turned tori are all diffeomorphic to $S^2 \times S^2$, and show that a family of homotopy 4-spheres constructed by Juhász and Powell are all diffeomorphic to $S^4$. |
| title | Branched covers of twist-roll spun knots and turned twisted tori |
| topic | Geometric Topology 57K45, 57K40 |
| url | https://arxiv.org/abs/2402.11706 |