Computing the Teichmüller polynomial (Q1687390)

Let \(M\) be a fibered hyperbolic 3-manifold. The integer points in the fibered cone \(R^+ \cdot F \subset H ^1 (M, R)\) over the fibered face \(F \) of the Thurston norm unit ball correspond to fibrations of \( M \) over the circle. If \(M \) is hyperbolic, the monodromy of each such fibration is a pseudo-Anosov class \([\psi]\) with stretch factor \(\lambda(\psi) > 1\). These stretch factors are packaged in the Teichmüller polynomial, defined in [\textit{C. T. McMullen}, Ann. Sci. Éc. Norm. Supér. (4) 33, No. 4, 519--560 (2000; Zbl 1013.57010)]. This is an element \( \Theta_F =\Sigma_{g\in G} a_ g g \) in the group ring \(Z[H _1 (M,Z)/\text{Torsion}]\), which is associated to the fibered face \(F\) and that is used to compute the stretch factor \(\lambda(\psi)\). The authors provide a general algorithm for computing the Teichmüller polynomial given a pseudo-Anosov mapping class obtained as a loop in a train track automaton. The author's algorithm derives all the relevant information on the topology of various fibers that belong to a fibered face.