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A pseudo-Anosov flow on a hyperbolic 3-manifold dynamically represents a faceF of the Thurston norm ball if the cone on F is dual to the cone spanned byhomology classes of closed orbits of the flow. Fried showed that for everyfibered face of the Thurston norm ball there is a unique, up to isotopy andreparametrization, flow which dynamically represents the face. Using veeringtriangulations we have found that there are non-fibered faces of the Thurstonnorm ball which are dynamically represented by multiple topologicallyinequivalent flows. This raises a question of how distinct flows representingthe same face are related.  We define combinatorial mutations of veering triangulations along surfacesthat they carry. We give sufficient and necessary conditions for the mutanttriangulation to be veering. After appropriate Dehn filling these veeringmutations correspond to transforming one 3-manifold M with a pseudo-Anosov flowtransverse to an embedded surface S into another 3-manifold admitting apseudo-Anosov flow transverse to a surface homeomorphic to S. We show that anon-fibered face of the Thurston norm ball can be dynamically represented bytwo distinct flows that differ by a veering mutation.

Mutations and faces of the Thurston norm ball dynamically represented by multiple distinct flows