# Rigidity versus flexibility for tight confoliations - Mathematics > Geometric Topology

Abstract: In \cite{confol} Y. Eliashberg and W. Thurston gave a definition of tightconfoliations. We give an example of a tight confoliation $\xi$ on $T^3$violating the Thurston-Bennequin inequalities. This answers a question from\cite{confol} negatively. Although the tightness of a confoliation does notimply the Thurston-Bennequin inequalities, it is still possible to proverestrictions on homotopy classes of plane fields which contain tightconfoliations.The failure of the Thurston-Bennequin inequalities for tight confoliations isdue to the presence of overtwisted stars. Overtwisted stars are particularconfigurations of Legendrian curves which bound a disc with finitely manypunctures on the boundary. We prove that the Thurston-Bennequin inequalitieshold for tight confoliations without overtwisted stars and that symplecticallyfillable confoliations do not admit overtwisted stars.

Author: T. Vogel

Source: https://arxiv.org/