The Legendrian knot complement problem

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.

Cosmetic contact surgeries along transverse knots and the knot complement problem

“…The proof of the transverse knot exterior theorem works now very similar as for topological or Legendrian knots. Compare [Ke16,Ke17].…”

Computing rotation and self-linking numbers in contact surgery diagrams

“…The knot L 0 is nullhomologous in M if and only if there is an integral solution a of the equation l = Qa (see [13]).…”

“…Let L 0 ⊂ S 3 \ L be an oriented knot in the complement of an oriented surgery link L. Using the notation from Section 2, we see that L 0 is rationally nullhomologous of order d in M = S 3 L (r) if and only if there is an integral solution a of the equation dl = Qa and d is the minimal natural number for which a solution exists (see [13]). Now assume that L and L 0 are Legendrian and L 0 is rationally nullhomologous of order d in M and fix Seifert surfaces Σ 0 , .…”

“…A natural extension is to consider Legendrian or transverse knots in contact surgery diagrams along Legendrian links and to compute their classical invariants in the surgered manifold. Starting with the work of Lisca, Ozsváth, Stipsicz and Szabó [14,Lemma 6.6] several results were obtained in that setting by Geiges and Onaran [10, Lemma 2], Conway [3,Lemma 6.4] and Kegel [13,Section 8].…”

“…In particular, t is the negative change of the Thurston-Bennequin number of L 0 in the surgery (cf. [3], [13]), i.e. we get…”