Non-loose unknots, overtwisted discs, and the contact mapping class group of $S^3$

TL;DR

This paper classifies non-loose Legendrian unknots in overtwisted contact structures on S^3, revealing their invariants, and uses these results to analyze the contact mapping class group and contactomorphism group actions.

Contribution

It provides a complete classification of non-loose Legendrian unknots in overtwisted S^3 and applies this to study the contact mapping class group and contactomorphism group.

Findings

Exactly two non-loose Legendrian unknots for each (n, ±(n-1)) with n>0

Only one overtwisted structure admits a non-loose unknot with specified invariants

The contact mapping class group depends on the contact structure

Abstract

We classify Legendrian unknots in overtwisted contact structures on $S^3$. 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 $S^3$ with Thurston-Bennequin invariant $n$ and rotation number $\pm(n-1)$. (Only one overtwisted contact structure on $S^3$ admits a non-loose unknot $K$ and the classical invariants have to be $\mathrm{tb}(K)=n$ and $\mathrm{rot}(K)=\pm(n-1)$ for $n>1$.) This can be used to prove two results attributed to Y.~Che\-kan\-ov: 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.