Let \\(M\\) be a three-manifold with\n nondegenerate Thurston norm \\(x\\) on its second homology.\n There is a partial dictionary between the co mbinatorics\n of the polyhedral unit ball of \\(x\\) and\n the topological features of \\(M\\). This dictionary is\n quite incomplete\, but its existing entries are tantalizing.\n
\n\n Currently\, most of the entries of this dictionary co ncern\n fibered faces of the unit ball. Thurston proved that these \n organize all fibrations of \\(M\\) over the circle. Fried and\n Mosher tell us more: for each fibered face \\(F\\) there is a\n (canonical) pseudo-Anosov flow whose Euler class computes the\n norm \\(x\\) in the cone over \\(F\\). Furthermore\, the flow\n "sees" certain least-complexity surfaces. Further work of\n Mosher shows that\, under certain conditions\, pseudo-Anosov\n flows can n aturally specify nonfibered faces of the unit ball.\n
\n\n After giving some of this background I will discuss results\n from my recent preprint (see link). I\n show that Agol's veer ing triangulations can be used to\n determine faces of Thurston nor m balls\, to compute the\n Thurston norm over those faces\, and to collate all isotopy\n classes of least-complexity surfaces over tho se faces. This\n analysis includes nonfibered faces.\n
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