The Thurston norm of 2-bridge link complements

Abstract: The Thurston norm is a seminorm on the second real homology group of a compact orientable 3-manifold. The unit ball of this norm is a convex polyhedron, whose shape's data (e.g. number of vertices, regularity) measures the complexity of the surfaces sitting in the ambient 3-manifold. Unfortunately, the Thurston norm is generally quite hard to compute, and a long-standing problem is to understand which polyhedra are realised as the unit balls of the Thurston norms of $3$-manifolds. We show that, when $M$ is the complement of a $2$-bridge link $L$ with components $\ell_1$ and $\ell_2$, the Thurston ball of $M$ has at most 8 faces. The proof of this result strongly relies on a description of essential surfaces in $2$-bridge link complements given by Floyd and Hatcher. Then, we exhibit norm-minimizing representatives for the integral classes of $H_2(M,\partial M)$ and use them to compare the complexity of the Thurston ball with the complexities of $L$ and of $M$. As an example, we show that all the vertices of the Thurston ball lie on the bisectors if and only if $M$ fibers over the circle with fiber a surface with boundary equal to a longitude of $\ell_1$ and some meridians of $\ell_2$. Finally, we use $2$-bridge links in satellite constructions to find $2$-component links whose complements in $S^3$ have Thurston balls with arbitrarily many vertices.