I am sorry but I am reposting this question because I wasn't logged in when I first asked it.
Ian Agol has produced a method to build an ideal layered triangulation of a hyperbolic 3-manifold which fibers over the circle with pseudo-Anosov monodromy of the fiber through periodic splitting of train tracks. Such triangulation is transverse-taut and veering.
On the converse, he proves that a layered triangulation of a manifold coming from a periodic sequence of Whitehead moves comes from a periodic train track splitting only if it is veering (see arxiv.org/pdf/1008.1606).
Both this construction and the notion of layered triangulation require to specify the fiber and the monodromy.
From Thurston's work, it is known that a hyperbolic 3-manifold $M$ which fibers over the circle fibers in many different ways, and in fact the fibers of the different fibrations are integral points of the cones over certain faces of the unit ball for the thurston norm in $H_2(M)$.
I was wondering how to compare the different layered triangulations of the same manifold that one can construct from the different fiberings. More precisely:
Are two veering layered triangulations built from two different fibrations whose fibers lie in the cone over a common fibering face for the thurston norm combinatorially the same? If this is true, are their taut structures the same? If not, how can we relate the taut structures?
Is there a way to compare triangulations built from fibers in different fibered faces for the thurston norm?
Thank you very much for your attention.