Two-bridge knots admit no purely cosmetic surgeries

Kazuhiro Ichihara; In Dae Jong; Thomas W. Mattman; Toshio Saito

doi:10.2140/agt.2021.21.2411

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arXiv · 1909.02340

Two-bridge knots admit no purely cosmetic surgeries

Kazuhiro Ichihara·In Dae Jong·Thomas W. Mattman·Toshio Saito

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Abstract

We show that two-bridge knots and alternating fibered knots admit no purely cosmetic surgeries, i.e., no pair of distinct Dehn surgeries on such a knot produce 3-manifolds that are homeomorphic as oriented manifolds. Our argument, based on a recent result by Hanselman, uses several invariants of knots or 3-manifolds; for knots, we study the signature and some finite type invariants, and for 3-manifolds, we deploy the $SL(2,\mathbb{C})$ Casson invariant.

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Kazuhiro Ichihara, In Dae Jong, Thomas W. Mattman, Toshio Saito. 2019-09-05. Two-bridge knots admit no purely cosmetic surgeries. https://doi.org/10.2140/agt.2021.21.2411

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