The taut polynomial and the Alexander polynomial

Anna Parlak

Published 2021-01-28Version 1

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.