The taut polynomial and the Alexander polynomial

TL;DR

This paper establishes a deep connection between the taut polynomial of veering triangulations and twisted Alexander polynomials, providing formulas relating them and extending McMullen's theorem to nonfibred 3-manifolds.

Contribution

It proves the equality of the taut polynomial and a twisted Alexander polynomial, and derives formulas linking the taut and classical Alexander polynomials in various cases.

Findings

Taut polynomial equals a certain twisted Alexander polynomial.

Formulas relate taut polynomial and Alexander polynomial depending on edge-orientability.

Extension of McMullen's theorem to nonfibred 3-manifolds.

Abstract

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.