Thurston norm and taut branched surfaces

Abstract:

Let $x$ denote the Thurston norm on ${H_2}(N;{\mathbf {R}})$, where $N$ is a closed, oriented, irreducible, atoroidal three-manifold $N$. U. Oertel defined a taut oriented branched surface to be a branched surface with the property that each surface it carries is incompressible and $x$-minimizing for the (nontrivial) homology class it represents. Given $\varphi$, a face of the $x$-unit sphere in ${H_2}(N;{\mathbf {R}})$, Oertel then asks: is there a taut oriented branched surface carrying surfaces representing every integral homology class projecting to $\varphi$? In this article, an example is constructed for which the answer is negative.