Computation of the Taut, the Veering and the Teichmüller Polynomials

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.

KEYWORDS:

• 3-manifolds
• veering triangulation
• taut polynomial
• veering polynomial
• Teichmüller polynomial
• fibration over the circle

Acknowledgements

I am grateful to Samuel Taylor for explaining to me his work on the veering polynomial during my visit at Temple University in July 2019, and subsequent conversations. I thank Saul Schleimer and Henry Segerman for their generous assistance in implementing the algorithms presented in this paper. This implementation is based on their Veering Census [9] and accompanying tools for computing with veering triangulations. I thank Mark Bell for answering my questions about flipper. I also thank the referees for many helpful suggestions.

Additional information

Funding

This work has been written during PhD studies of the author at the University of Warwick under the supervision of Saul Schleimer. It was supported by The Engineering and Physical Sciences Research Council (EPSRC) under grant EP/N509796/1 studentship 1936817.