Abstract
Given a closed, irreducible, orientable 3-manifold M, let x denote the Thurston norm on H 2 ( M ; R ) {H_2}(M;R) . Suppose g, h, and f are three homology classes of H 2 ( M ; Z ) {H_2}(M;Z) carried by a single face of the x-unit sphere in H 2 ( M ; R ) {H_2}(M;R) . In this paper it is shown that there exists a taut, oriented branched surface carrying representatives of g and h and a semi-taut oriented branched surface carrying representatives of all three homology classes.
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