A note on orientation-reversing distance one surgeries on non-null-homologous knots

Author:

Ito Tetsuya

Abstract

We show that there are no distance one surgeries on non-null-homologous knots in M M that yield M -M ( M M with opposite orientation) if M M is a 3-manifold obtained by a Dehn surgery on a knot K K in S 3 S^{3} , such that the order of its first homology is divisible by 9 9 but is not divisible by 27 27 .

As an application, we show several knots, including the ( 2 , 9 ) (2,9) torus knot, do not have chirally cosmetic bandings. This simplifies the proof of a result first proven by Yang that the ( 2 , k ) (2,k) torus knot ( k > 1 ) (k>1) has a chirally cosmetic banding if and only if k = 5 k=5 .

Funder

Japan Society for the Promotion of Science

Publisher

American Mathematical Society (AMS)

Reference13 articles.

1. The lens space realization problem;Greene, Joshua Evan;Ann. of Math. (2),2013

2. Applications of the Casson-Walker invariant to the knot complement and the cosmetic crossing conjectures;Ito, Tetsuya;Geom. Dedicata,2022

3. Distance one lens space fillings and band surgery on the trefoil knot;Lidman, Tye;Algebr. Geom. Topol.,2019

4. Signatures, Heegaard Floer correction terms and quasi-alternating links;Lisca, Paolo;Proc. Amer. Math. Soc.,2015

5. Chiral smoothings of knots;Livingston, Charles;Proc. Edinb. Math. Soc. (2),2020

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