  Landry, Minsky and Taylor defined the taut polynomial of a veering
triangulation. Its specialisations generalise the Teichmuller polynomial of a
fibred face of the Thurston norm ball. We prove that the taut polynomial of a
veering triangulation is equal to a certain twisted Alexander polynomial of the
underlying manifold. Then we give formulas relating the taut polynomial and the
untwisted Alexander polynomial. There are two formulas; one holds when the
maximal free abelian cover of a veering triangulation is edge-orientable,
another holds when it is not edge-orientable.
  Furthermore, we consider 3-manifolds obtained by Dehn filling a veering
triangulation. In this case we give a formula that relates the specialisation
of the taut polynomial under the Dehn filling and the Alexander polynomial of
the Dehn-filled manifold. This extends a theorem of McMullen connecting the
Teichmuller polynomial and the Alexander polynomial to the nonfibred setting,
and improves it in the fibred case. We also prove a sufficient and necessary
condition for the existence of an orientable fibred class in the cone over a
fibred face of the Thurston norm ball.
