Abstract
Landry, Minsky and Taylor (LMT) introduced two polynomial invariants of veering triangulations—the taut polynomial and the veering polynomial. Here, we consider a pair of taut polynomials associated to one veering triangulation, the upper and the lower one, and analogously the upper and lower veering polynomials. We prove that the upper and lower taut polynomials are equal. In contrast, the upper and lower veering polynomials of the same veering triangulation may differ by more than a unit. We give algorithms to compute all these invariants. LMT related the Teichmüller polynomial of a fibered face of the Thurston norm ball with the taut polynomial of the associated layered veering triangulation. We use this result to give an algorithm to compute the Teichmüller polynomial of any fibered face of the Thurston norm ball.
Highlights
Transverse taut veering triangulations were introduced by Ian Agol as a way to canonically triangulate certain pseudo-Anosov mapping tori [1]
We consider a pair of taut polynomials associated to one veering triangulation, the upper and the lower one, and analogously the upper and lower veering polynomials
That together with TautPolynomial make up the algorithm TeichmüllerPolynomial. We prove that this algorithm correctly computes the Teichmüller polynomial of any fibered face of the Thurston norm ball
Summary
Transverse taut veering triangulations were introduced by Ian Agol as a way to canonically triangulate certain pseudo-Anosov mapping tori [1]. The computation of the taut polynomial requires calculating only linearly many minors of a matrix, instead of exponentially many (Proposition 5.6) Using this we give algorithm TautPolynomial and prove that it correctly computes the taut polynomial of a veering triangulation in Proposition 5.8. Since there are 51,766 layered veering triangulations in the Veering Census, the implementation of the algorithm TautPolynomial alone expands the existing collection of computed Teichmüller polynomials from a couple of examples [2, 4, 13, 14] to almost 52 thousand of examples. Further generalization to fiber-parallel Dehn fillings of layered veering triangulations, given in the algorithm TeichmüllerPolynomial, yields even more computable examples
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