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Vogel, Thomas (2018): Non-loose unknots, overtwisted discs, and the contact mapping class group of S (3). In: Geometric and Functional Analysis, Vol. 28, No. 1: pp. 228-288

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Abstract

We classify Legendrian unknots in overtwisted contact structures on S (3). In particular, we show that up to contact isotopy for every pair with n > 0 there are exactly two oriented non-loose Legendrian unknots in S (3) with Thurston-Bennequin invariant n and rotation number . (Only one overtwisted contact structure on S (3) admits a non-loose unknot K and the classical invariants have to be tb(K) = n and 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 S (3) depends on the contact structure. The second result is that the identity component of the contactomorphism group of an overtwisted contact structure on S (3) does not always act transitively on the set of boundaries of overtwisted discs.

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