# Cosmetic contact surgeries along transverse knots and the knot complement problem

## TL;DR

This paper investigates cosmetic contact surgeries on transverse knots in the standard contact 3-sphere, showing that only trivial surgeries occur for most knots and establishing that transverse knots are determined by their exteriors, with some exceptions.

## Contribution

It proves the non-existence of non-trivial cosmetic contact surgeries for all transverse knots except the transverse unknot with self-linking -1, and relates knot exteriors to knot determination.

## Key findings

- Non-trivial cosmetic contact surgeries are excluded for all transverse knots except the transverse unknot with self-linking -1.
- Transverse knots in the standard contact 3-sphere are uniquely determined by the contactomorphism type of their exteriors.
- Counterexamples are provided for transverse links, showing limitations of the main result.

## Abstract

We study cosmetic contact surgeries along transverse knots in the standard contact 3-sphere, i.e. contact surgeries that yield again the standard contact 3-sphere. The main result is that we can exclude non-trivial cosmetic contact surgeries along all transverse knots not isotopic to the transverse unknot with self-linking number -1. As a corollary it follows that every transverse knot in the standard contact 3-sphere is determined by the contactomorphism type of its exteriors. Moreover, we give counterexamples to this for transverse links in the standard contact 3-sphere.