Topologically Trivial Legendrian Knots

This paper deals with topologically trivial Legendrian knots in tight and overtwisted contact 3-manifolds. The first parts (Sections 1-3) contain a thorough exposition of the proof of the classification of topologically trivial Legendrian knots (i.e. Legendrian knots bounding embedded 2-disks) in tight contact 3-manifolds (Theorem 1.7), and, in particular, in the standard contact S. These parts were essentially written more than 10 years ago, but only a short version [EF], without the detailed proofs, was published. In that paper we also briefly discussed Legendrian knots in overtwisted contact 3-manifolds. The final part of the present paper (Section 4) contains a more systematic discussion of the overtwisted case. In [EF] Legendrian knots in overtwisted manifolds were divided into two classes: exceptional, i.e. those with tight complement, and the complementary class of loose ones. Loose knots can be coarsely, i.e. up to a global coorientation preserving contact diffeomorphism, classified (see Section 4 and also [D2]) using the classification of overtwisted contact structures from [E5]. This implies, in particular, the Legendrian isotopy classification of topologically trivial loose knots with tb < 0 according to the values of the invariants tb, r. On the other hand, Giroux-Honda’s classification of tight contact structures on solid tori (see [Gi3, Ho1, Ho2]) allows us to completely coarsely classify topologically trivial exceptional knots in S. In particular, we show the latter exist for only one overtwisted contact structure on S. Since the paper [EF], several new techniques (notably the Giroux–Honda method of convex surfaces, dividing curves and Legendrian bypasses) were