An algorithm to compute the Teichmüller polynomial from matrices

Abstract

In their precedent work, the authors constructed closed oriented hyperbolic surfaces with pseudo-Anosov homeomorphisms from certain class of integral matrices. In this paper, we present a very simple algorithm to compute the Teichmüller polynomial corresponding to those surface homeomorphisms by first constructing an invariant track whose first homology group can be naturally identified with the first homology group of the surface, and computing its Alexander polynomial.

Computation of the taut, the veering and the Teichmüller polynomials

“…Let (T , α) be a transverse taut triangulation of M . Since T is endowed with a compatible taut structure, we can view the 2-skeleton T (2) of T as a 2-dimensional complex with a well-defined tangent space everywhere, including along its 1-skeleton. Thus T (2) determines a branched surface (without vertices) in M .…”

“…Since T is endowed with a compatible taut structure, we can view the 2-skeleton T (2) of T as a 2-dimensional complex with a well-defined tangent space everywhere, including along its 1-skeleton. Thus T (2) determines a branched surface (without vertices) in M . It is called the horizontal branched surface and denoted by B [27, Subsection 2.12].…”

“…In [19] Lanneau and Valdez present an algorithm to compute the Teichmüller polynomial of punctured disk bundles. The authors of [2] compute the Teichmüller polynomial of oddblock surface bundles. In [4] Billet and Lechti cover the case of alternatingsign Coxeter links.…”