Mixed t\^ete-\`a-t\^ete twists as monodromies associated with holomorphic function germs

TL;DR

This paper characterizes mixed tête-à-tête twists as pseudo-periodic automorphisms related to monodromies of isolated complex surface singularities, generalizing earlier models with t extasciitilde{}te- extasciitilde{}te graphs.

Contribution

It introduces mixed tête-à-tête twists and characterizes them as automorphisms with specific periodic properties, linking them to monodromies of complex surface singularities.

Findings

Mixed tête-à-tête twists are characterized as automorphisms with a power being a product of Dehn twists.

The class of tête-à-tête twists coincides with monodromies of reduced function germs on isolated surface singularities.

Provides a topological model for monodromies in complex surface singularity theory.

Abstract

T\^ete-\`a-t\^ete graphs were introduced by N. A'Campo in 2010 with the goal of modeling the monodromy of isolated plane curves. Mixed t\^ete-\`a-t\^ete graphs provide a generalization which define mixed t\^ete-\`a-t\^ete twists, which are pseudo-periodic automorphisms on surfaces. We characterize the mixed t\^ete-\`a-t\^ete twists as those pseudo-periodic automorphisms that have a power which is a product of right-handed Dehn twists around disjoint simple closed curves, including all boundary components. It follows that the class of t\^ete-\`a-t\^ete twists coincides with that of monodromies associated with reduced function germs on isolated complex surface singularities.