Sutured manifold hierarchies and the Thurston nom
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918459927953408 |
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| author | Cigna, Alessandro V. |
| author_facet | Cigna, Alessandro V. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_19611 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sutured manifold hierarchies and the Thurston nom Cigna, Alessandro V. Geometric Topology Primary: 57K31, Secondary: 55M25, 57R30, 57K10 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. |
| title | Sutured manifold hierarchies and the Thurston nom |
| topic | Geometric Topology Primary: 57K31, Secondary: 55M25, 57R30, 57K10 |
| url | https://arxiv.org/abs/2604.19611 |