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

TL;DR

This paper introduces and characterizes t extsuperscript{et extsuperscript{a}- extsuperscript{t extsuperscript{e}}}t extsuperscript{e} twists and graphs, providing new tools to represent and analyze mapping class groups and monodromies of plane curve singularities.

Contribution

It defines and characterizes t extsuperscript{et extsuperscript{a}- extsuperscript{t extsuperscript{e}}}t extsuperscript{e} twists, generalizes their applicability, and introduces mixed t extsuperscript{et extsuperscript{a}- extsuperscript{t extsuperscript{e}}}t extsuperscript{e} twists to represent monodromies.

Findings

Characterized mapping classes represented by t extsuperscript{et extsuperscript{a}- extsuperscript{t extsuperscript{e}}}t extsuperscript{e} twists.

Generalized the notion to boundary-free periodic classes.

Proved that mixed t extsuperscript{et extsuperscript{a}- extsuperscript{t extsuperscript{e}}}t extsuperscript{e} twists include monodromies of irreducible plane curve singularities.

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.