Non-loose unknots, overtwisted discs, and the contact mapping class group of S³

الأصل · EN

We classify Legendrian unknots in overtwisted contact structures on S³. In particular, we show that up to contact isotopy for every pair (n,±(n-1)) with n>0 there are exactly two oriented non-loose Legendrian unknots in S³ with Thurston-Bennequin invariant n and rotation number ±(n-1). (Only one overtwisted contact structure on S³ admits a non-loose unknot K and the classical invariants have to be tb(K)=n and rot(K)=±(n-1) for n>1.) This can be used to prove two results attributed to Y. Chekanov: The first implies that the contact mapping class group of an overtwisted contact structure on S³ depends on the contact structure. The second result is that the identity component of the contactomorphism group of an overtwisted contact structure on S³ does not always act transitively on the set of boundaries of overtwisted discs.