Cables of the figure-eight knot via real Frøyshov invariants
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912980535345152 |
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| author | Kang, Sungkyung Park, JungHwan Taniguchi, Masaki |
| author_facet | Kang, Sungkyung Park, JungHwan Taniguchi, Masaki |
| contents | We prove that the $(2n,1)$-cable of the figure-eight knot is not smoothly slice when $n$ is odd, by using the real Seiberg-Witten Frøyshov invariant of Konno-Miyazawa-Taniguchi. For the computation, we develop an $O(2)$-equivariant version of the lattice homotopy type, originally introduced by Dai-Sasahira-Stoffregen. This enables us to compute the real Seiberg-Witten Floer homotopy type for a certain class of knots. Additionally, we present some computations of Miyazawa's real framed Seiberg-Witten invariant for 2-knots. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_09295 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cables of the figure-eight knot via real Frøyshov invariants Kang, Sungkyung Park, JungHwan Taniguchi, Masaki Geometric Topology 57K10, 57K41 We prove that the $(2n,1)$-cable of the figure-eight knot is not smoothly slice when $n$ is odd, by using the real Seiberg-Witten Frøyshov invariant of Konno-Miyazawa-Taniguchi. For the computation, we develop an $O(2)$-equivariant version of the lattice homotopy type, originally introduced by Dai-Sasahira-Stoffregen. This enables us to compute the real Seiberg-Witten Floer homotopy type for a certain class of knots. Additionally, we present some computations of Miyazawa's real framed Seiberg-Witten invariant for 2-knots. |
| title | Cables of the figure-eight knot via real Frøyshov invariants |
| topic | Geometric Topology 57K10, 57K41 |
| url | https://arxiv.org/abs/2405.09295 |