T\^ete-\`a-t\^ete twists, monodromies and representation of elements of Mapping Class Group

Abstract

We study monodromies of plane curve singularities and pseudo-periodic homeomorphisms of oriented surfaces with boundary, following an original idea of the first author: t\^ete-\`a-t\^ete graphs and twists. We completely characterize mapping classes that can be represented by t\^ete-\`a-t\^ete twists, and generalize the notion to be able to represent any class of the mapping class group relative to the boundary which is boundary-free periodic. This improves previous work on the subject by C. Graf. Furthermore, we introduce the class of mixed t\^ete-\`a-t\^ete graphs and twists, and prove that mixed t\^ete-\`a-t\^ete twists contain monodromies of irreducible plane curve singularities. In a sequel paper, the fourth author and B. Sigurdsson have extended this to the reducible case.