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 h-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 h-principle for a subclass of loose Legendrians is also derived from the main result. In general, the method allows us to deduce an h-principle for overtwisted disks that are fixed near the boundary in an arbitrary overtwisted contact 3-manifold.