  Landry, Minsky and Taylor introduced a polynomial invariant of veering
triangulations called the taut polynomial. We prove that when a veering
triangulation is edge-orientable then its taut polynomial is equal to the
Alexander polynomial of the underlying manifold. For triangulations which are
not edge-orientable we give a sufficient condition for the equality between the
support of the taut polynomial and that of the Alexander polynomial. We also
consider Dehn fillings of 3-manifolds equipped with a veering triangulation. In
this case we compare the image of the taut polynomial under a Dehn filling and
the Alexander polynomial of the Dehn-filled manifold.
  As an application we extend the results of McMullen which relate the
Teichm\"uller polynomial with the Alexander polynomial. Using this we find
first cohomology classes corresponding to fibrations whose monodromy determines
orientable invariant laminations in the fibre.
