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

Anna Parlak

Published 2023-03-30Version 1

A pseudo-Anosov flow on a hyperbolic 3-manifold dynamically represents a face F of the Thurston norm ball if the cone on F is dual to the cone spanned by homology classes of closed orbits of the flow. Fried showed that for every fibered face of the Thurston norm ball there is a unique, up to isotopy and reparametrization, flow which dynamically represents the face. Using veering triangulations we have found that there are non-fibered faces of the Thurston norm ball which are dynamically represented by multiple topologically inequivalent flows. This raises a question of how distinct flows representing the same face are related. We define combinatorial mutations of veering triangulations along surfaces that they carry. We give sufficient and necessary conditions for the mutant triangulation to be veering. After appropriate Dehn filling these veering mutations correspond to transforming one 3-manifold M with a pseudo-Anosov flow transverse to an embedded surface S into another 3-manifold admitting a pseudo-Anosov flow transverse to a surface homeomorphic to S. We show that a non-fibered face of the Thurston norm ball can be dynamically represented by two distinct flows that differ by a veering mutation.