  We define the Thurston-Bennequin polytope of a two-component link as the
convex hull of all pairs of integers that arise as framings of a Legendrian
representative. The main result of this paper is a description of the
Thurston-Bennequin polytope for two-bridge links. As an application, we
construct non-quasipositive surfaces in $\R^3$ all whose sub-annuli are
quasipositive.
