Families of cosmetic surgeries

TL;DR

This paper constructs infinite examples of cosmetic surgeries on hyperbolic knots and manifolds, challenging previous conjectures and providing insights into their occurrence.

Contribution

It provides the first infinite families of chirally and purely cosmetic surgeries on hyperbolic knots and manifolds, disproving longstanding conjectures.

Findings

Constructed infinite families of chirally cosmetic surgeries on chiral hyperbolic knots.

Constructed infinite families of purely cosmetic surgeries on hyperbolic manifolds with multiple cusps.

Disproved conjectures suggesting such phenomena do not occur.

Abstract

We construct infinite families of chirally cosmetic surgeries on chiral hyperbolic knots and purely cosmetic surgeries on hyperbolic manifolds with multiple cusps, disproving conjectures that these phenomena do not appear, including Problem 1.12(d) in the K3 problem list. We also give some hints regarding why chirally cosmetic surgeries appear to be more common than purely cosmetic surgeries on $1$-cusped manifolds.