Computation of the taut, the veering and the Teichmüller polynomials

Anna Parlak

Published 2020-09-28Version 1

Landry, Minsky and Taylor [LMT] introduced two polynomial invariants of veering triangulations - the taut polynomial and the veering polynomial. We give algorithms to compute these invariants. In their definition [LMT] use only the upper track of the veering triangulation, while we consider both the upper and the lower track. We prove that the lower and the upper taut polynomials are equal. However, we show that there are veering triangulations whose lower and upper veering polynomials are different. [LMT] proved that when a veering triangulation is layered, the taut polynomial is equal to the Teichm\"uller polynomial of the corresponding fibred face of the Thurston norm ball. Thus the algorithm presented in this paper computes the Teichm\"uller polynomial of a fully-punctured fibred face. We generalise this algorithm to the case of fibred faces which are not fully-punctured.