Cosmetic surgery on satellite knots

TL;DR

This paper proves that if any knot in the 3-sphere admits purely cosmetic surgeries, then a hyperbolic knot with this property must also exist, narrowing the search for such knots.

Contribution

It establishes that purely cosmetic surgeries, if they occur, must be found among hyperbolic knots, providing a significant restriction on potential counterexamples.

Findings

Purely cosmetic surgeries imply the existence of hyperbolic examples.

Hyperbolic knots are the only candidates for cosmetic surgeries if such surgeries exist.

The result narrows the scope of knots to consider for cosmetic surgery phenomena.

Abstract

We show that if there exists a knot in $S^3$ that admits purely cosmetic surgeries, then there exists a hyperbolic one with this property.