Tête-à-tête twists, monodromies and representation of elements of mapping class group (Q2152457)

The paper studies monodromies of plane curve singularities and pseudo-periodic homeomorphisms of oriented surfaces with boundary using tête-à-tête graphs and twists. The (Max) Dehn twist is defined as follows. Given an embedded copy \(\alpha\) of the circle in the interior of an oriented surface \(S\), a mapping class \(D_{\alpha}\) of the surface \(S\) is defined. This class has for every open subset \(U\) that contains \(\alpha\) a representative \(\delta _U\) with support in \(U\), that leaves \(\alpha\) invariant and whose restriction to \(\alpha\) has order 2. A tête-à-tête twist is a generalisation of the classical Dehn twist. The authors of the present paper introduce the class of mixed tête-à-tête graphs and twists, and prove that mixed tête-à-tête twists contain the monodromies of irreducible plane curve singularities. In a sequel paper, the fourth author and \textit{B. Sigurðsson} [Geom. Dedicata 210, 43--64 (2021; Zbl 1458.32024)] have extended this to the reducible case.