# Cosmetic surgeries and the Poincare homology sphere

## TL;DR

This paper proves that in the Poincare homology sphere, homotopically essential knots cannot have purely cosmetic surgeries, highlighting a topological restriction on such surgeries in this specific 3-manifold.

## Contribution

It establishes a new restriction on cosmetic surgeries for knots in the Poincare homology sphere, a notable result in 3-manifold topology.

## Key findings

- Homotopically essential knots in the Poincare homology sphere do not admit purely cosmetic surgeries.
- Provides a topological obstruction to cosmetic surgeries in this specific manifold.
- Advances understanding of knot surgeries in homology spheres.

## Abstract

In this short note, we prove that if a knot in the Poincare homology sphere is homotopically essential, then it does not admit any purely cosmetic surgeries.