Agol introduced veering triangulations of mapping tori,
whose combinatorics are canonically associated to the pseudo-Anosov
monodromy. In unpublished work, Guéritaud and Agol generalise an
alternative construction to any closed manifold equipped with a
pseudo-Anosov flow without perfect fits.

Schleimer and I build the reverse map: from a transverse veering
triangulation we canonically construct a dynamic pair in the sense of
Mosher, which gives a combinatorial version of a pseudo-Anosov flow
without perfect fits. Along the way, we construct a canonical circular
ordering of the cusps of the universal cover of a veering
triangulation.

I will also talk about work with Giannopolous and Schleimer building a
census of transverse veering triangulations. The current census lists
all transverse veering triangulations with up to 16 tetrahedra, of
which there are 87,047. The number approximately doubles with each
added tetrahedron.