overtwistedness

Overtwistedness is a property of a contact structure on a 3-manifold (and, in a generalized sense, on higher-dimensional manifolds) indicating a high level of flexibility in the geometry. A contact structure is a nowhere integrable plane field ξ, locally described as the kernel of a 1-form α with α ∧ dα ≠ 0. The structure is overtwisted if the manifold contains an embedded disk D^2, called an overtwisted disk, whose boundary ∂D is Legendrian and along which the characteristic foliation on D exhibits a particular pattern with an interior singularity. If no such disk exists, the structure is called tight.

In dimension three, overtwisted and tight structures form a dichotomy with different implications for classification. Tight contact structures are rigid and sensitive to topology, while overtwisted ones are flexible. Eliashberg proved in 1989 that on a closed oriented 3-manifold, overtwisted contact structures up to isotopy correspond exactly to homotopy classes of oriented plane fields. This allows a complete classification in the overtwisted case. More recently, Borman, Eliashberg, and Giroux established a general h-principle for overtwisted contact structures, showing that in all dimensions they are determined by their formal data and that any two homotopic almost contact structures can be realized by overtwisted contact structures that are isotopic.

In higher dimensions, the concept extends via the presence of higher-dimensional analogues of overtwisted disks (often called plastikstufes). The BEG framework demonstrates that overtwisted higher-dimensional contact structures satisfy flexibility properties analogous to the 3-dimensional case, while tight structures retain rigidity.

Common examples include the standard tight contact structures on R^3 and S^3; performing a Lutz twist along a transverse knot yields an overtwisted structure.