Thurston norm and taut branched surfaces

Abstract

Let x x denote the Thurston norm on H 2 ( N ; R ) {H_2}(N;{\mathbf {R}}) , where N N is a closed, oriented, irreducible, atoroidal three-manifold N 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 x -minimizing for the (nontrivial) homology class it represents. Given φ \varphi , a face of the x x -unit sphere in H 2 ( N ; R ) {H_2}(N;{\mathbf {R}}) , Oertel then asks: is there a taut oriented branched surface carrying surfaces representing every integral homology class projecting to φ \varphi ? In this article, an example is constructed for which the answer is negative.