Given a knot exterior, a natural problem is to determine whether different Dehn fillings on its boundary may yield homeomorphic oriented manifolds. If this situation happens, the homeomorphic manifolds are said to be obtained from each other via purely cosmetic Dehn surgery.
Recent work of J. Hanselman [“Heegaard Floer homology and cosmetic surgeries in \(S^3\)”, Preprint, arXiv:1906.06773, to appear in J. Eur. Math. Soc.] provides necessary conditions for slopes on knots in the \(3\)-sphere to give rise to cosmetic surgeries. Building on his results, the authors analyse in detail three classes of knots, that is \(2\)-bridge knots, alternating fibred knots, and alternating pretzel knots, and prove that they admit no purely cosmetic surgery.
The strategy of the proof is to exploit different types of knot invariants to establish which knots in these classes fulfill Hanselman’s conditions. For the two latter classes, the authors show that the knots never fulfill the conditions. The situation is more involved for \(2\)-bridge knots, for which it is necessary to distinguish the manifolds obtained by surgery on potential cosmetic slopes by using \(3\)-manifolds invariants, like Casson’s \(SL(2,{\mathbb C})\) one.
In the study of \(2\)-bridge knots, results on cosmetic surgery by K. Ichihara and Z. Wu [Commun. Anal. Geom. 27, No. 5, 1087–1104 (2019; Zbl 1436.57005)] are also used.