Thurston norm gluing along an incompressible torus

In an irreducible three-manifold with incompressible boundary, a minimizing surface can be cut along a separating incompressible torus after removing inessential intersection circles. The essential pieces have no disk or sphere components, so their ${\chi }_{-}$ costs add. Minimizing independently in the pieces and gluing their matching boundary curves gives the corresponding upper bound. This allows Thurston norm calculations for satellite knots.