= Thurston norm gluing along an incompressible torus
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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>.
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