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

Abstract

We consider a surface bundle over the circle, the so-called magic manifold M. We determine homology classes whose minimal representatives are genus 0 fiber surfaces for M, and describe their monodromies by braids. Among those classes whose representatives have n punctures for each n, we decide which one realizes the minimal entropy. We show that for each n ≥ 9 (resp. n = 3, 4,5, 7, 8), there exists a pseudo-Anosov homeomorphism Φn: Dn → Dn with the smallest known entropy (resp. the smallest entropy) which occurs as the monodromy on an n-punctured disk fiber for the Dehn filling of M. A pseudo-Anosov homeomorphism Φ6: D6 → D6 with the smallest entropy occurs as the monodromy on a 6-punctured disk fiber for M.