# $PSL_2(\mathbb{C})$-character varieties and Seifert fibered cosmetic surgeries

## TL;DR

This paper investigates the constraints on cosmetic surgeries on knots in rational homology spheres, using $PSL_2(bC)$-character varieties to establish bounds on slopes yielding identical small Seifert manifolds.

## Contribution

It provides a sharp bound on the number of slopes producing the same small Seifert manifold under certain representation-theoretic conditions.

## Key findings

- Bound on the number of slopes for cosmetic surgeries
- Application of $PSL_2(bC)$-character variety theory
- Results applicable to rational homology spheres

## Abstract

We study small Seifert possibly chiral cosmetic surgeries on not necessarily null-homologous knot in rational homology spheres. Using $PSL_2(\mathbb{C})$-character variety theory we give a sharp bound on the number of slopes producing the same small Seifert manifold if the ambient manifold satisfies some representation theoretic conditions.