Legendrian Knots in Overtwisted Contact Structures on S3

Abstract

We study the problem of classifying Legendrian knots in overtwisted contact structures on S3. The question is whether topologically isotopic Legendrian knots have to be Legendrian isotopic if they have equal values of the well-known invariants rot and tb. We give positive answer in the case that there is an overtwisted disc intersecting none of the knots and we construct an example of a knot intersecting each overtwisted disc (this provides a counterexample to the conjecture of Eliashberg). Our proof needs some results on the structure of the group of contactomorphisms of S3. We divide the subgroup Cont+(S3, ξ) of coorientation-preserving contactomorphisms for an overtwisted contact distribution ξ into two classes.