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(2, ) 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
Mathematical Subject Classification 2010
Primary: 57M27
Secondary: 57M25
References
Publication
Received: 5 September 2019
Revised: 17 January 2020
Accepted: 4 October 2020
Published: 31 October 2021
Authors
Kazuhiro Ichihara
Department of Mathematics
College of Humanities and Sciences
Nihon University
Tokyo
Japan
http://www.math.chs.nihon-u.ac.jp/~ichihara/index.html
In Dae Jong
Department of Mathematics
Kindai University
Osaka
Japan
Thomas W Mattman
Department of Mathematics and Statistics
California State University at Chico
Chico, CA
United States
Toshio Saito
Department of Mathematics
Joetsu University of Education
Joetsu
Japan