In this thesis we study the taut polynomial of a veering triangulation, defined by Landry, Minsky and Taylor [LMT]. We introduce the notion of edge-orientability and 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 that are not edge-orientable, we give a sufficient condition for equality between the support of the taut polynomial and that of the Alexander polynomial.

We also 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.

Using the results of [LMT] we extend the theorem of McMullen which relates the Alexander polynomial of a manifold and the Teichmï¿œuller polynomial of a fibred face of its Thurston norm ball. We interpret the obtained result in terms of existence of orientable fibred classes in the corresponding fibred cone.

The computational part of the thesis includes algorithms to compute the taut polynomial, the upper veering polynomial, and their specialisations. Using this and the results of [LMT] we give an algorithm to compute the Teichmï¿œuller polynomial of any fibred face of the Thurston norm ball. We also present an algorithm to find the face of the Thurston norm ball determined by a veering triangulation. We use implementations of these algorithms to compute examples illustrating the theoretical results of the thesis.