Let N g denote a closed nonorientable surface of genus g. For g ≥ 2 the mapping class group M(N g ) is generated by Dehn twists and one crosscap slide (Y -homeomorphism) or by Dehn twists and a crosscap transposition. Margalit and Schleimer observed that Dehn twists on orientable surfaces have nontrivial roots. We give necessary and sufficient conditions for the existence of roots of crosscap slides and crosscap transpositions.

Landry, Minsky, and Taylor introduced an invariant of veering triangulations called the taut polynomial. Via a connection between veering triangulations and pseudo-Anosov flows, it generalizes the Teichmüller polynomial of a fibered face of the Thurston norm ball to (some) non-fibered faces. We construct a sequence of veering triangulations, with the number of tetrahedra tending to infinity, whose taut polynomials vanish. These veering triangulations encode non-circular Anosov flows transverse to tori. Contents 1. Introduction 1 2. Veering triangulations, the taut polynomial, and vertical surgeries 3 3. Sequence of veering triangulations with a vanishing taut polynomial 12 References 24 Appendix A. The Jacobian of (3.7) 25

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)