Landry, Minsky and Taylor defined the taut polynomial of a veering triangulation. Its specialisations generalise the Teichmüller 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. Thus, the Teichmüller polynomials are just specialisations of twisted Alexander polynomials. We also give formulae relating the taut polynomial and the untwisted Alexander polynomial. There are two formulae, depending on whether the maximal free abelian cover of a veering triangulation is edge-orientable or not. Furthermore, we consider 3-manifolds obtained by Dehn filling a veering triangulation. In this case, we give formulae that relate the specialisation of the taut polynomial under a Dehn filling and the Alexander polynomial of the Dehn-filled manifold. This extends a theorem of McMullen connecting the Teichmüller polynomial and the Alexander polynomial to the non-fibred 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.

中文翻译：

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拉紧多项式和亚历山大多项式

Landry、Minsky 和 ​​Taylor 定义了转向三角剖分的拉紧多项式。它的专业化概括了 Thurston 范数球的纤维面的 Teichmüller 多项式。我们证明转向三角剖分的张紧多项式等于基础流形的某个扭曲亚历山大多项式。因此，Teichmüller 多项式只是扭曲的亚历山大多项式的特化。我们还给出了有关拉紧多项式和未扭曲的亚历山大多项式的公式。有两个公式，取决于转向三角剖分的最大自由阿贝尔覆盖是否是边可定向的。此外，我们考虑由 Dehn 填充转向三角剖分获得的 3-流形。在这种情况下，我们给出的公式将 Dehn 填充下的拉紧多项式的特化与 Dehn 填充流形的亚历山大多项式联系起来。这扩展了将 Teichmüller 多项式和亚历山大多项式连接到非纤维设置的 McMullen 定理，并在纤维情况下对其进行了改进。我们还证明了在 Thurston 范数球的纤维面上的锥体中存在可定向纤维类的充分必要条件。