Sutured manifold hierarchies and the Thurston nom

Abstract

Classical work of Thurston and Gabai shows that finitely many taut sutured manifold hierarchies determine the Thurston norm of a compact oriented irreducible $3$-manifold with toroidal boundary. We give an explicit procedure to extract this information from such hierarchies. This is achieved via the maw dual graph construction, which can be incorporated into a general method for computing the Thurston norm of a manifold. As an application, we compute the Thurston norm of the exterior of all alternating and some nonalternating pretzel links with three components. Using these computations, we give a negative answer to a question of Baker--Taylor. Moreover, we show that if a nonseparating surface $S$ in a Haken manifold $M$ with toroidal boundary is disjoint from a boundary torus, then the class $[S] \in H_2(M,\partial M)$ does not lie in the interior of a top-dimensional cone of the Thurston norm. In particular, if two components $\ell_i$ and $\ell_j$ of a nonsplit link have zero linking number, then neither represents a class in an open top-dimensional cone of the Thurston norm ball of the link exterior.