Computation of the taut, the veering and the Teichm\"uller polynomials

TL;DR

This paper develops algorithms to compute the taut, veering, and Teichmüller polynomials associated with veering triangulations, extending previous work by considering both upper and lower tracks and establishing new relationships among these invariants.

Contribution

It introduces algorithms for computing the taut, veering, and Teichmüller polynomials, including the novel consideration of both upper and lower tracks and the proof of their relationships.

Findings

Lower and upper taut polynomials are equal.

Lower and upper veering polynomials can differ.

Algorithm for computing the Teichmüller polynomial of fibred faces.

Abstract

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] related the Teichm\"uller polynomial of a fibred 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 Teichm\"uller polynomial of any fibred face of the Thurston norm ball.