The overtwisted contact structure on S^3 with Hopf invariant -1 admits a contractible positive loop of contactomorphisms, making it the first known non-orderable overtwisted contact manifold.
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A non-orderable overtwisted contact structure on the sphere
The overtwisted contact structure on S^3 with Hopf invariant -1 admits a contractible positive loop of contactomorphisms, making it the first known non-orderable overtwisted contact manifold.