The taut polynomial and the Alexander polynomial

Abstract

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 Dehnfilled 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.M S C 2 0 2 0 57K31 (primary), 37E30 (secondary)

Computation of the Taut, the Veering and the Teichmüller Polynomials

“…Remark 2.2. Taut triangulations are often called transverse taut triangulations; see for instance [16,18,21].…”

“…such that ω(γ) = 1 if and only if γ lifts to a loop in the orientable double cover of B; see [18,Section 4]. Let π : π 1 (M ) → H be the natural projection arising from the abelianization and killing the torsion of H 1 (M ; Z).…”

“…19. By [18,Theorem 5.7], the Alexander polynomials of manifolds underlying ( V k ) k≥0 also vanish. The Alexander polynomials of manifolds underlying (V k ) k≥0 do not vanish.…”

“…In fact, starting from the same veering triangulation V 0 one can construct a different sequence of veering triangulations with vanishing taut polynomials by choosing a different vertical surgery curve; see Remark 3. 18.…”

“…Let V be a finite veering triangulation of a 3-manifold M . In [18,Theorem 5.7] the author showed that the taut polynomial of V is equal to the twisted Alexander polynomial ∆ ω⊗π M , where ω : π 1 (M ) → {−1, 1} is a certain homomorphism determined by V (see Section 2.4.2) and π : π 1 (M ) → H 1 (M ; Z)/torsion is the abelianization and torsion-killing projection. Since there are only |H 1 (M ; Z/2)| homomorphisms π 1 (M ) → Z/2, it follows that for any 3-manifold M there are only finitely many candidates for the taut polynomial of a veering triangulation of M .…”