Thurston norms of tunnel number-one manifolds

Abstract

The Thurston norm of a three-manifold measures the complexity of surfaces representing two-dimensional homology classes. We study the possible unit balls of Thurston norms of three-manifolds [Formula: see text] with [Formula: see text], and whose fundamental groups admit presentations with two generators and one relator. We show that even among this special class, there are three-manifolds such that the unit ball of the Thurston norm has arbitrarily many faces.