Sutured manifold hierarchies and the Thurston nom

TL;DR

This paper presents an explicit method to compute the Thurston norm of 3-manifolds using sutured manifold hierarchies and applies it to link exteriors, providing new insights and answering existing questions.

Contribution

It introduces a procedure to extract Thurston norm information from sutured manifold hierarchies via the maw dual graph construction, enhancing computational techniques.

Findings

Computed Thurston norm for all alternating and some nonalternating pretzel links with three components.

Provided a negative answer to a question of Baker--Taylor regarding the Thurston norm.

Showed that certain nonseparating surfaces do not lie in the interior of a top-dimensional cone of the Thurston norm.

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.