Purely cosmetic surgeries and pretzel knots

TL;DR

This paper proves that for pretzel knots, different Dehn surgeries always produce distinct three-manifolds, confirming the purely cosmetic surgery conjecture for this class of knots.

Contribution

It establishes that all pretzel knots satisfy the purely cosmetic surgery conjecture, a significant step in understanding Dehn surgeries on knots.

Findings

All pretzel knots satisfy the cosmetic surgery conjecture.

Different slopes on pretzel knots lead to non-homeomorphic three-manifolds.

Confirms the conjecture for a broad class of knots.

Abstract

We show that all pretzel knots satisfy the (purely) cosmetic surgery conjecture, i.e. Dehn surgeries with different slopes along a pretzel knot provide different oriented three-manifolds.