[Paper Review] The minimum dilatation of pseudo-Anosov 5-braids

Ji-Young Ham, Won Taek Song|ArXiv.org|Jun 15, 2005

Geometric and Algebraic Topology18 references4 citations

TL;DR

This paper determines the minimum dilatation of pseudo-Anosov 5-braids by analyzing train track representatives and transition matrices, proving that the minimal dilatation is the largest root $\lambda_5 \approx 1.72208$ of the polynomial $x^4 - x^3 - x^2 - x + 1$, achieved by the braid $\sigma_1\sigma_2\sigma_3\sigma_4\sigma_1\sigma_2$. The result is obtained through a computational search using folding automata and spectral analysis of Perron-Frobenius matrices.

ABSTRACT

The minimum dilatation of pseudo-Anosov 5-braids is shown to be the largest zero $λ_5 \approx 1.72208$ of $x^4 - x^3 - x^2 - x + 1$ which is attained by $σ_1σ_2σ_3σ_4σ_1σ_2$.

Motivation & Objective

• To determine the infimum of dilatations among all pseudo-Anosov 5-braids.
• To identify the specific braid achieving this minimal dilatation.
• To develop and apply a computational framework based on folding automata and transition matrices for enumerating candidate pseudo-Anosov braids.
• To verify that the candidate braid satisfies the necessary spectral and topological conditions for minimality.

Proposed method

• Construct a folding automaton from embedded train tracks on the 5-punctured sphere, using elementary folding maps and isomorphisms.
• Label edges of train tracks and compute transition matrices for all possible closed paths in the automaton.
• Apply spectral analysis to transition matrices, focusing on Perron-Frobenius matrices with spectral radius less than 3 to filter candidates.
• Use pruning rules based on row sums of transition matrices to eliminate paths with spectral radius exceeding 3.
• Implement a search algorithm (fbrmin) that avoids forbidden subwords and tracks closed paths with dilatation below a given bound.
• Verify the minimal dilatation by computing the largest eigenvalue of the transition matrix associated with the candidate braid.

Experimental results

Research questions

• RQ1 What is the smallest possible dilatation among all pseudo-Anosov 5-braids?
• RQ2 Which specific braid realizes this minimal dilatation?
• RQ3 Can the minimal dilatation be computed algorithmically using train track and matrix transition methods?
• RQ4 How can computational complexity be reduced during the search for minimal dilatation braids?

Key findings

• The minimum dilatation of pseudo-Anosov 5-braids is $\lambda_5 \approx 1.72208$, the largest real root of the polynomial $x^4 - x^3 - x^2 - x + 1$.
• This minimal dilatation is achieved by the braid $\sigma_1\sigma_2\sigma_3\sigma_4\sigma_1\sigma_2$.
• The transition matrix associated with this braid is Perron-Frobenius and has spectral radius equal to $\lambda_5$.
• The computation required approximately 1000 seconds and 150 MB of memory on a 2.40 GHz machine, testing around 85,000 matrices.
• The search strategy effectively prunes paths with row sums exceeding 3 in the transition matrix, as such matrices necessarily have spectral radius greater than 3.
• The method is extendable to higher braid indices, with the authors expecting 6-braids to be computationally feasible on standard personal computers.