Cosmetic contact surgeries along transverse knots and the knot complement problem
Geometric Topology
2026-02-10 v2 Symplectic Geometry
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.
Keywords
Cite
@article{arxiv.1703.05596,
title = {Cosmetic contact surgeries along transverse knots and the knot complement problem},
author = {Marc Kegel},
journal= {arXiv preprint arXiv:1703.05596},
year = {2026}
}
Comments
15 pages, 7 figures; v2: Conditions on the size of the tubular neighborhoods are removed from the results; Examples 5.2 is corrected; Small changes in the presentation