Examples of pseudo-Anosov homeomorphisms

Abstract:

We generalize a construction in knot theory to construct a large family $\mathcal {G}\mathcal {R} = \cup GR(\mathcal {P})$ of mapping classes of a surface of genus $g$ and one boundary component, where $\mathcal {P}$ runs over some finite index set. We exhibit explicitly the set $\mathcal {G}{\mathcal {R}^{\ast }} \subset \mathcal {G}\mathcal {R}$ that consists of pseudo-Anosov maps, find the map that realizes the smallest dilatation in $\mathcal {G}{\mathcal {R}^{\ast }}$, and for every $\mathcal {P}$, we give a set of defining relations for $GR(\mathcal {P})$.