When incompressible tori meet essential laminations

We show that there is only one obstruction to isotoping an essential lamination L in a 3-manifold M containing an essential torus T so that when both L and M are split open along T, the resulting lamination L|T is essential in the split open manifold M |T. This is the existence of a cylindrical component in L. This generalizes previous results relating taut foliations and essential tori. 0. Introduction. Roussarie and Thurston [Ro], [Th] showed that given a taut foliation F and an incompressible torus T in a 3-manifold M, the torus may be isotoped so that it is either transverse to the leaves of the foliation, or is equal to one of the leaves of the foliation. This result was known to be false in th