Rigidity versus flexibility for tight confoliations

TL;DR

This paper demonstrates that tight confoliations can violate Thurston-Bennequin inequalities due to overtwisted stars, but these inequalities hold if such configurations are absent, highlighting nuanced differences in confoliation tightness.

Contribution

It provides a counterexample of tight confoliations violating Thurston-Bennequin inequalities and establishes conditions under which these inequalities are preserved.

Findings

Counterexample of tight confoliation violating inequalities

Overtwisted stars cause failure of inequalities

Fillable confoliations do not have overtwisted stars

Abstract

In \cite{confol} Y. Eliashberg and W. Thurston gave a definition of tight confoliations. 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 not imply the Thurston-Bennequin inequalities, it is still possible to prove restrictions on homotopy classes of plane fields which contain tight confoliations. The failure of the Thurston-Bennequin inequalities for tight confoliations is due to the presence of overtwisted stars. Overtwisted stars are particular configurations of Legendrian curves which bound a disc with finitely many punctures on the boundary. We prove that the Thurston-Bennequin inequalities hold for tight confoliations without overtwisted stars and that symplectically fillable confoliations do not admit overtwisted stars.