Standard contact structure (ℝ³)
α = dz − y dx ⟹ ξ_std = ker(α)
Planes tilt in x-direction as y increases.
α ∧ dα = dx ∧ dy ∧ dz ≠ 0 (contact condition)
Tight: no overtwisted disk
Overtwisted structure
α_OT = cos(r)dz + r·sin(r)dθ (polar)
Contains overtwisted disk: D where ξ|∂D = T(∂D)
Classified by homotopy class of plane field (Eliashberg 1989)
Reeb vector field: ι_R α=1, ι_R dα=0