The Thurston norm of 2-bridge link complements
Alessandro V. Cigna · Journal of Knot Theory and Its Ramifications · 2025
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 [Formula: see text] and [Formula: see text], 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 in [W. Floyd and A. Hatcher, The space of incompressible surfaces in a 2-bridge link complement, Trans. Am. Math. Soc. 305(2) (1988) 575–599]. Then, we exhibit norm-minimizing representatives for the integral classes of [Formula: see text] 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 [Formula: see text] and some meridians of [Formula: see text]. Finally, we use 2-bridge links in satellite constructions to find 2-component links whose complements in [Formula: see text] have Thurston balls with arbitrarily many vertices.