The Legendrian knot complement problem
Geometric Topology
2026-02-10 v2 Symplectic Geometry
Abstract
We prove that every Legendrian knot in the tight contact structure of the 3-sphere is determined by the contactomorphism type of its exterior. Moreover, by giving counterexamples we show this to be not true for Legendrian links in the tight 3-sphere. On the way a new user-friendly formula for computing the Thurston-Bennequin invariant of a Legendrian knot in a surgery diagram is given.
Cite
@article{arxiv.1604.05196,
title = {The Legendrian knot complement problem},
author = {Marc Kegel},
journal= {arXiv preprint arXiv:1604.05196},
year = {2026}
}
Comments
32 pages, 14 figures; V2: Completly rewritten following the referee's advice. The results on non-equivalent Legendrian knots in lens spaces with contactomorphic exteriors contained a gap and are removed