Arbitrarily large veering triangulations with a vanishing taut polynomial

Abstract

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. The conclusion is that the size of a veering triangulation does not correlate with the ‘complexity’ of its taut polynomial. Furthermore, our result shows that the taut polynomials of veering triangulations representing non-fibered faces of the Thurston norm ball behave differently than the Teichmüller polynomials.