Transverse knots have unique standard neighborhoods, non-loose Legendrian knots satisfy a structure theorem, and tight contact manifolds contain only finitely many transverse knots up to isotopy, all following from a general Legendrian destabilization result that also produces a 3-manifold with unkn
Etnyre and Bülent Tosun.Low dimensional contact geometry
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Neighborhoods of transverse knots and destabilizations
Transverse knots have unique standard neighborhoods, non-loose Legendrian knots satisfy a structure theorem, and tight contact manifolds contain only finitely many transverse knots up to isotopy, all following from a general Legendrian destabilization result that also produces a 3-manifold with unkn