Computation of the Taut, the Veering and the Teichmüller Polynomials

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. These veering triangulations encode non-circular Anosov flows transverse to tori. Contents 1. Introduction 1 2. Veering triangulations, the taut polynomial, and vertical surgeries 3 3. Sequence of veering triangulations with a vanishing taut polynomial 12 References 24 Appendix A. The Jacobian of (3.7) 25

The taut polynomial and the Alexander polynomial

“…Pick the orientation on the edges of $$\mathcal {V}$$ determined by the transverse orientation on $$\tau ^L$$. Note that for any triangle $$f \in F$$ the lower large edge of $$f$$ and the upper large edge of $$f$$ are different edges of $$f$$, but both are of the same colour as $$f$$ [32, Lemma 2.1]. Figure 6 presents the lower and upper tracks in a veering triangle.…”

“…An algorithm to compute the Teichmüller polynomial using Lemma 7.1 is given in [32, section 8]. As mentioned in the introduction, Corollary 7.2 implies a much faster algorithm to compute the Teichmüller polynomial via Fox calculus.…”

“…An algorithm to compute the taut polynomial from its original definition was previously known [32] and implemented [33]. Given a veering triangulation with $$n$$ tetrahedra it required finding the greatest common divisor of the maximal minors of a certain matrix of dimension $$n\times (n+1)$$ with coefficients in $$\mathbb {Z}[H_M ]$$ [32, Corollary 5.7]. Unfortunately, computing determinants of matrices with entries in $$\mathbb {Z}[H_M ]$$ is slow, thus having to compute as many as $$n+1$$ determinants of $$n\times n$$ matrices was an obvious drawback of this algorithm.…”

“…Thus, the matrices one needs to work with have lower dimension. Computation of the taut polynomial using Fox calculus has been implemented [33] and it indeed proved to be much faster than the original computation.…”

Flows, growth rates, and the veering polynomial

“…This connects to our previous work [LMT20] introducing the veering polynomial, relating it to the Teichmüller polynomial, and laying the combinatorial groundwork for what is done here (although we emphasize that this paper can be read independently of the previous). Also, Parlak has recently introduced and implemented algorithms to compute the veering polynomial and its relatives [Par21a], and demonstrated a connection with the Alexander polynomial [Par21b], thereby generalizing the work of McMullen on the Teichmüller polynomial [McM00].…”