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March 2018 Cosmetic surgery and the $SL(2,\mathbb{C})$ Casson invariant for two-bridge knots
Kazuhiro Ichihara, Toshio Saito
Hiroshima Math. J. 48(1): 21-37 (March 2018). DOI: 10.32917/hmj/1520478021

Abstract

We consider the cosmetic surgery problem for two-bridge knots in the 3-sphere. We first verify by using previously known results that all the two-bridge knots of at most $9$ crossings admit no purely cosmetic surgery pairs except for the knot $9_{27}$. Then we show that any two-bridge knot corresponding to the continued fraction $[0, 2x, 2, -2x, 2x, 2, -2x]$ for a positive integer $x$ admits no cosmetic surgery pairs yielding homology 3-spheres, where $9_{27}$ appears when $x = 1$. Our advantage to prove this is using the $SL(2,\mathbb{C})$ Casson invariant.

Citation

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Kazuhiro Ichihara. Toshio Saito. "Cosmetic surgery and the $SL(2,\mathbb{C})$ Casson invariant for two-bridge knots." Hiroshima Math. J. 48 (1) 21 - 37, March 2018. https://doi.org/10.32917/hmj/1520478021

Information

Received: 1 November 2016; Revised: 18 April 2017; Published: March 2018
First available in Project Euclid: 8 March 2018

zbMATH: 06901785
MathSciNet: MR3771998
Digital Object Identifier: 10.32917/hmj/1520478021

Subjects:
Primary: 57M50
Secondary: 57M25, 57M27, 57N10

Rights: Copyright © 2018 Hiroshima University, Mathematics Program

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Vol.48 • No. 1 • March 2018
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