A remark on the finiteness of purely cosmetic surgeries

Ito, Tetsuya

doi:10.2140/agt.2023.23.2213

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A remark on the finiteness of purely cosmetic surgeries

Tetsuya Ito

Algebraic & Geometric Topology 23 (2023) 2213–2219

DOI: 10.2140/agt.2023.23.2213

Abstract

By estimating the knot Floer thickness in terms of the genus and the braid index, we show that a knot $K$ in $S^{3}$ does not admit purely cosmetic surgery whenever $g (K) \geq \frac{3}{2} b (K)$ , where $g (K)$ and $b (K)$ denote the genus and the braid index, respectively. In particular, this establishes the finiteness of purely cosmetic surgeries; for a fixed $b$ , all but finitely many knots with braid index $b$ satisfies the cosmetic surgery conjecture.

Keywords

cosmetic surgery, knot Floer thickness, braid index

Mathematical Subject Classification

Primary: 57K10

Secondary: 57K30

References

Publication

Received: 25 April 2021

Revised: 24 November 2021

Accepted: 17 January 2022

Published: 25 July 2023

Authors

Tetsuya Ito
Department of Mathematics Graduate School of Science Kyoto University Kyoto Japan

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Volume 23, issue 5 (2023)