The Thurston norm of 2-bridge link complements

TL;DR

This paper investigates the shape and complexity of the Thurston norm unit ball for 2-bridge link complements, establishing bounds on the number of faces and exploring conditions for fibered structures.

Contribution

It provides a bound of at most 8 faces for the Thurston ball of 2-bridge link complements and characterizes when vertices lie on bisectors, advancing understanding of Thurston norm realizations.

Findings

Thurston ball of 2-bridge link complements has at most 8 faces.

Vertices of the Thurston ball lie on bisectors if and only if the manifold fibers over the circle.

Constructs examples of links with Thurston balls having arbitrarily many vertices.

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.