Cosmetic contact surgeries along transverse knots and the knot complement problem
arXiv:1703.05596 · doi:10.1016/j.topol.2019.01.020
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.
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