General t\^ete-\`a-t\^ete graphs and Seifert manifolds

TL;DR

This paper introduces general t extasciitilde{}a-t extasciitilde{}e graphs, extending previous models to encompass all periodic mapping classes, and provides algorithms linking Seifert manifolds and these graphs.

Contribution

It defines general t extasciitilde{}a-t extasciitilde{}e graphs and proves they model all periodic mapping classes, along with algorithms connecting Seifert manifolds and these graphs.

Findings

General t extasciitilde{}a-t extasciitilde{}e graphs model all periodic mapping classes.

Algorithms convert between Seifert manifolds with horizontal surfaces and t extasciitilde{}a-t extasciitilde{}e graphs.

Abstract

T\^ete-\`a-t\^ete graphs and relative t\^ete-\`a-t\^ete graphs were introduced by N. A'Campo in 2010 to model monodromies of isolated plane curves. By recent workof Fdez de Bobadilla, Pe Pereira and the author, they provide a way of modeling the periodic mapping classes that leave some boundary component invariant. In this work we introduce the notion of general t\^ete-\`a-t\^ete graph and prove that they model all periodic mapping classes. We also describe algorithms that take a Seifert manifold and a horizontal surface and return a t\^ete-\`a-t\^ete graph and vice versa.