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.

APPENDIX A. Odd-block matrices with entries bigger than 1

In the main text of the paper, the authors provided an algorithm to compute the Teichmüller polynomial for odd-block surfaces, and the odd-block surfaces are constructed as described in Sect. 2 following [3]. One of the limitations of the construction of odd-block surfaces is that the given odd-block matrix is assumed to have only 0, 1 entries. The main purpose of this assumption is to guarantee the uniqueness of the corresponding piecewise-linear map $h_M$. See Fig. 5 for an example of an odd-block matrix with entries bigger than 1 where $h_M$ is not unique.

Fig. 5

The piecewise-linear map $h_M$ is not uniquely determined in the above example

On the other hand, for any given odd-block matrix, one can obtain an odd-block matrix with $\{0,1\}$-entries with essentially the same information. More precisely, we prove the following theorem.

Theorem 4

Let M be an $n \times n$ non-singular, aperiodic, odd-block, nonnegative integral matrix. For each choice of the piecewise-linear map $h_M$, then there exists an aperiodic, odd-block matrix N with only $\{0, 1\}$-entries such that $h_N$ coincides with $h_M$. Furthermore, the leading eigenvalue of N is the leading eigenvalue of M.

Let M be a matrix as in Theorem 4. For instance, one can consider an example shown in the left part of Fig. 6. We show that there is a canonical way to convert M into another aperiodic odd-block matrix ${\overline{M}}$ with only 0 and 1 entries, having the same leading eigenvalue (say $\lambda$).

Fig. 6

Left An odd-block matrix M with a corresponding piecewise-linear map $h_M$. Right A matrix obtained by splitting the matrix on the left

Let v be the $L^1$-normalized eigenvector of $M^T$ for the leading eigenvalue. As we did in Sect. 2, we get a partition $P=\lbrace x_0 , x_1, \ldots , x_n \rbrace$ of [0, 1] so that $v_i=x_i-x_{i-1}$ for each $i=1,\ldots ,n$. Let’s consider the $n\times n$ grid diagram on $[0,1]\times [0,1]$ generated by the partition P. Each (i, j) box corresponds to $M_{ij}$ so that M is flipped upside down. Because M is odd-block, it is always possible to draw a graph of piecewise-linear map $h_M$ so that the number of line segments of $h_M$ in each box is the same with the corresponding entry of M. As we draw on the grid diagram, the slopes of $h_M$ are either $\lambda$ or $-\lambda$. There may be few possible graphs with that property, but one can choose any of them. Note that the conversion depends on the choice of the graph.

Now we are in a position to convert M. Let ${\overline{P}}=\lbrace y_0 , y_1, \ldots , y_n \rbrace$ be the union of P and the set of all critical points of $h_M$. Then since the post-critical set of $h_M$ is contained in P, $h_M({\overline{P}})\subset {\overline{P}}$ and so ${\overline{P}}$ is invariant under $h_M$. The extended incident matrix, say ${\overline{M}}$, of $h_M$ associated with ${\overline{P}}$ is the desired converted matrix. See the right part of Fig. 6 for a resulting matrix ${\overline{M}}$ of this process. Note that ${\overline{M}}$ is inevitably singular because of duplicated rows.

Each entry of ${\overline{M}}$ is 0 or 1. Since ${\overline{P}}$ includes all critical points of $h_M$, the vector $w=(w_1,\ldots ,w_m)$, $w_i=y_i-y_{i-1}$ for each $i=1,\ldots ,m$ is the $L^1$-normalized eigenvector of ${\overline{M}}^T$ for $\lambda$. This is because the equation $({\overline{M}}^Tw)_i=\lambda w_i$ simply represents the length relation between $[y_{i-1},y_i]$ and $h_M([y_{i-1},y_i])$.

To prove ${\overline{M}}$ is aperiodic, we use following fact: For positive integer p, i-th column of ${\overline{M}}^p$ is positive if and only if $h_M^p([y_{i-1},y_i])=[0,1]$. Since M is aperiodic and $h_M([y_{i-1},y_i])=[x_j, x_k]$ for some $j<k$, $h_M^p([y_{i-1},y_i])$ eventually covers [0, 1] as p grows, ${\overline{M}}$ is aperiodic. Finally, from the Perron-Frobenius theorem, we conclude that $\lambda$ is the leading eigenvalue of ${\overline{M}}$ as it is the associated eigenvalue of the positive eigenvector w of the aperiodic matrix ${\overline{M}}$.