Pseudo-Anosov braids with small entropy and the magic $3$-manifold

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.