A pseudo-Anosov flow on a hyperbolic 3-manifold dynamically represents a top-dimensional
face of the Thurston
norm ball if the cone on
is dual to the cone spanned by the 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 nonfibered faces of the Thurston
norm ball which are dynamically represented by multiple topologically inequivalent
flows. This raises the 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
with a pseudo-Anosov flow transverse to an embedded surface
into
another 3-manifold admitting a pseudo-Anosov flow transverse to a surface homeomorphic
to
.
We show that a nonfibered face of the Thurston norm ball can be dynamically
represented by two distinct flows that differ by a veering mutation. Furthermore, one
of the discussed pairs of homeomorphic veering mutants can be used to construct
counterexamples to the classification theorem of Anosov flows on Bonatti–Langevin
manifolds published in the 90s.
