Topologically trivial Legendrian knots

Abstract

The first part of this paper contains 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. These techniques were never published in detail when the classification result was announced over ten years ago. The final part of the present paper contains a systematic discussion of Legendrian knots in overtwisted contact manifolds, and in particular, gives the coarse classification (i.e., classification up to a global contactomorphism) of topologically trivial exceptional Legendrian knots in overtwisted contact $S^3$ according to the values of the invariants tb, r. We show, moreover, that such knots only occur for one of the infinitely many overtwisted contact structures on $S^3$. We remark that our tight classification result also implies that any topologically trivial loose Legendrian knots with same value of (tb, r) in an overtwisted contact 3-manifold are in fact Legendrian isotopic if $tb < 0$.