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

Abstract

We classify Legendrian unknots in overtwisted contact structures on S3. In particular, we show that up to contact isotopy for every pair $${(n,\pm(n-1))}$$ with n > 0 there are exactly two oriented non-loose Legendrian unknots in S3 with Thurston–Bennequin invariant n and rotation number $${\pm(n-1)}$$ . (Only one overtwisted contact structure on S3 admits a non-loose unknot K and the classical invariants have to be tb(K) = n and $${{\rm rot}(K)=\pm(n-1)}$$ 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.