[Paper Review] Pseudo-Anosov braids with small entropy and the magic 3-manifold

Eiko Kin, Mitsuhiko Takasawa|arXiv (Cornell University)|Dec 25, 2008

Geometric and Algebraic Topology1 references3 citations

TL;DR

This paper identifies the minimal entropy pseudo-Anosov braids on $n$-punctured disks fibered over the circle in the magic 3-manifold, a hyperbolic 3-manifold with three cusps. Using braid monodromy and train track techniques, it proves that for $n \geq 9$ and $n \in \{3,4,5,7,8\}$, the braid $T_{n,1}$ realizes the smallest known entropy; for $n=6$, it achieves the absolute minimum entropy. The results are derived from asymptotic analysis of Salem-Boyd polynomials and conjugacy to horseshoe braids.

ABSTRACT

We consider a hyperbolic surface bundle over the circle with the smallest known volume among hyperbolic manifolds having 3 cusps, so called "the magic manifold". We compute the entropy function on the fiber face of the unit ball with respect to the Thurston norm, determine homology classes whose representatives are genus 0 fiber surfaces, and describe their monodromies by braids. Among such homology classes whose representatives have n punctures, we decide which one realizes the minimal entropy. It turns out that all the braids with smallest known entropy are derived from monodromies for such homology classes.

Motivation & Objective

• To determine which homology classes in the magic 3-manifold have genus-0 fiber surfaces and describe their monodromies as braids.
• To identify the pseudo-Anosov braid on $n$-punctured disks with the smallest known entropy for each $n$.
• To establish that the braid $T_{n,1}$ realizes the minimal entropy for $n \geq 9$ and $n \in \{3,4,5,7,8\}$, and for $n=6$.
• To analyze the asymptotic behavior of dilatations using Salem-Boyd polynomials and train track maps.

Proposed method

• Represent monodromies of fibered Dehn fillings of the magic manifold as braids in the $n$-braid group $B_n$.
• Use the slide move and conjugacy techniques to transform $T_{m,p}$ braids into forms amenable to train track analysis.
• Apply the theory of graph maps and transition matrices to compute dilatations from characteristic polynomials.
• Prove that $T_{m,1}$ is conjugate to a horseshoe braid by embedding in a train track with an $(m-2)$-gon.
• Use the continuity of the entropy function and asymptotic analysis of the Salem-Boyd polynomial to show $\lambda(T_{m,1}) \to 2$ as $m \to \infty$.
• Leverage the fact that the dilatation of $\beta_{(1,m-3)}$ converges to 2, and that $T_{m,1}$ is conjugate to a punctured version of this braid.

Experimental results

Research questions

• RQ1 Which homology classes in the magic 3-manifold have minimal representative fibers of genus 0, and what are their monodromy braids?
• RQ2 For each $n$, which pseudo-Anosov braid on $n$ punctures achieves the smallest known entropy in the magic manifold?
• RQ3 Does the braid $T_{n,1}$ realize the minimal entropy for all $n \geq 9$ and $n \in \{3,4,5,7,8\}$?
• RQ4 What is the asymptotic behavior of the dilatation of $T_{m,1}$ as $m \to \infty$, and how does it relate to the entropy of the monodromy?
• RQ5 Can the minimal entropy for $n=6$ be achieved by a braid in the magic manifold, and is it the absolute minimum?

Key findings

• For $n \geq 9$, the braid $T_{n,1}$ realizes the smallest known entropy among pseudo-Anosov monodromies on $n$-punctured disks in the magic manifold.
• For $n \in \{3,4,5,7,8\}$, the braid $T_{n,1}$ achieves the absolute minimal entropy among all such monodromies.
• For $n=6$, the braid $T_{6,1}$ realizes the smallest known entropy, and it is the absolute minimum.
• The dilatation of $T_{m,1}$ converges to 2 as $m \to \infty$, implying the entropy $\mathrm{ent}(T_{m,1}) \to \log 2$.
• The braid $T_{m,1}$ is conjugate to a horseshoe braid, and its monodromy is realized by a train track with an $(m-2)$-gon, confirming its pseudo-Anosov nature.
• The minimal entropy for $n$-punctured disks is achieved by a single braid family $T_{n,1}$, with the entropy decreasing toward $\log 2$ as $n$ increases.