Purely cosmetic surgeries and Casson--Walker--Lescop invariants
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912963183509504 |
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| author | Ichihara, Kazuhiro Jong, In Dae Tsutsumi, Yasuyoshi |
| author_facet | Ichihara, Kazuhiro Jong, In Dae Tsutsumi, Yasuyoshi |
| contents | Using the rational surgery formula for the Casson--Walker--Lescop invariant of links in the $3$-sphere, we show that any null-homologous knot in a rational homology sphere admits at most two pairs of integral purely cosmetic surgeries. We also present constraints for null-homologous knots in certain $3$-manifolds with the first Betti number one or two to admit purely cosmetic surgeries. As another application, we show that, for a null-homologous knot in some $3$-manifolds, including $S^2 \times S^1$, there are at most two knots which are inequivalent to the given one, but whose exteriors are orientation-preservingly homeomorphic to that of the given one. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_11720 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Purely cosmetic surgeries and Casson--Walker--Lescop invariants Ichihara, Kazuhiro Jong, In Dae Tsutsumi, Yasuyoshi Geometric Topology Primary 57K31, Secondary 57K10 Using the rational surgery formula for the Casson--Walker--Lescop invariant of links in the $3$-sphere, we show that any null-homologous knot in a rational homology sphere admits at most two pairs of integral purely cosmetic surgeries. We also present constraints for null-homologous knots in certain $3$-manifolds with the first Betti number one or two to admit purely cosmetic surgeries. As another application, we show that, for a null-homologous knot in some $3$-manifolds, including $S^2 \times S^1$, there are at most two knots which are inequivalent to the given one, but whose exteriors are orientation-preservingly homeomorphic to that of the given one. |
| title | Purely cosmetic surgeries and Casson--Walker--Lescop invariants |
| topic | Geometric Topology Primary 57K31, Secondary 57K10 |
| url | https://arxiv.org/abs/2603.11720 |