Tête-à-tête 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.