Sutured manifold hierarchies and the Thurston norm

Abstract

By the classical work of Thurston and Gabai, the Thurston norm of any compact oriented irreducible $3$-manifold with toroidal boundary is determined by finitely many taut sutured manifold hierarchies. 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 non-alternating pretzel links with three components. Using these computations, we give a negative answer to a question of Baker--Taylor. Namely, we exhibit an infinite family of manifolds, and an infinite family of knots in each manifold, such that the wrapping number associated with these knots is not a seminorm on the respective second real homology group. Next, we show that if a taut surface in a link exterior is disjoint from a boundary torus and does not represent a corner of the Thurston norm ball, then the corresponding link component is algebraically split from the rest of the link. Moreover, the surface remains norm-minimising under every non-longitudinal Dehn filling on that boundary component and, under an assumption of atoroidality, is a leaf of a taut foliation transverse to the filling core.