Strongly overtwisted contact 3‐manifolds

We prove the existence of a subclass of overtwisted contact structures, called strongly overtwisted, on a 3‐manifold that satisfy a complete ‐principle without prescribing the contact structures over any subset of the 3‐manifold. As a consequence, the homotopy type of the space of overtwisted disk embeddings into a strongly overtwisted contact 3‐manifold is determined. A complete ‐principle for a subclass of loose Legendrians is also derived from the main result. In general, the method allows us to deduce an ‐principle for overtwisted disks that are fixed near the boundary in an arbitrary overtwisted contact 3‐manifold.