  Recently, Ian Agol introduced a class of "veering" ideal triangulations for
mapping tori of pseudo-Anosov homeomorphisms of surfaces punctured along the
singular points. These triangulations have very special combinatorial
properties, and Agol asked if these are "geometric", i.e. realised in the
complete hyperbolic metric with all tetrahedra positively oriented. This paper
describes a computer program Veering, building on the program Trains by Toby
Hall, for generating these triangulations starting from a description of the
homeomorphism as a product of Dehn twists. Using this we obtain the first
examples of non-geometric veering triangulations; the smallest example we have
found is a triangulation with 13 tetrahedra.
