# An algorithm to compute the Teichmueller polynomial from matrices

## TL;DR

This paper introduces a straightforward algorithm to compute the Teichmueller polynomial for hyperbolic surface homeomorphisms derived from integral matrices, utilizing invariant tracks and Alexander polynomials.

## Contribution

The authors present a novel, simple algorithm that links invariant tracks and Alexander polynomials to efficiently compute the Teichmueller polynomial.

## Key findings

- Algorithm successfully computes Teichmueller polynomials from matrices.
- Method simplifies previous computational approaches.
- Applicable to surfaces with pseudo-Anosov homeomorphisms.

## 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 Teichmueller 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.