#### Volume 21, issue 5 (2021)

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Two-bridge knots admit no purely cosmetic surgeries

### Kazuhiro Ichihara, In Dae Jong, Thomas W Mattman and Toshio Saito

Algebraic & Geometric Topology 21 (2021) 2411–2424
##### Abstract

We show that two-bridge knots and alternating fibered knots admit no purely cosmetic surgeries, ie 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 $SL\left(2,ℂ\right)$ Casson invariant.

 Dedicated to Professor Hitoshi Murakami on his 60th birthday
##### Keywords
alternating knot, fibered knot, two-bridge knot, pretzel knot, cosmetic surgery, signature, finite type invariant, $\mathrm{SL}(2,C)$ Casson invariant
Primary: 57M27
Secondary: 57M25