  Landry, Minsky, and Taylor introduced an invariant of veering triangulations
called the taut polynomial. Via a connection between veering triangulations and
pseudo-Anosov flows, it generalizes the Teichm\"uller polynomial of a fibered
face of the Thurston norm ball to (some) non-fibered faces. We construct a
sequence of veering triangulations, with the number of tetrahedra tending to
infinity, whose taut polynomials vanish. These veering triangulations encode
non-circular Anosov flows transverse to tori.
