T\^ete-\`a-t\^ete graphs and twists

TL;DR

This thesis explores t extasciitilde{}ete- extasciitilde{}a-t extasciitilde{}ete twists, a combinatorial tool for mapping classes on surfaces, revealing their role in periodic classes, fiber surfaces, and the mapping class group.

Contribution

It introduces and analyzes t extasciitilde{}ete- extasciitilde{}a-t extasciitilde{}ete twists, showing their significance in describing periodic mapping classes and providing criteria for fibered Seifert surfaces.

Findings

T extasciitilde{}ete- extasciitilde{}a-t extasciitilde{}ete twists describe all periodic mapping classes.

A stronger version of Wiman's 4g+2 theorem is established for surfaces with boundary.

A criterion is provided to determine if a Seifert surface is a fiber surface.

Abstract

This is a PhD thesis in low-dimensional topology. Its main purpose is to examine so-called t\^ete-\`a-t\^ete twists. Those were defined by A'Campo and give an easy combinatorial description of certain mapping classes on surfaces with boundary. Whereas the well-known Dehn twists are twists around a simple closed curve, t\^ete-\`a-t\^ete twists are twists around a graph. It is shown that t\^ete-\`a-t\^ete twists describe all the (freely) periodic mapping classes. This leads, among other things, to a stronger version of Wiman's 4g+2 theorem from 1895 for surfaces with boundary. On closed surfaces, some t\^ete-\`a-t\^ete twists can be used to generate the mapping class group. Another main result is a simple criterion to decide whether a Seifert surface of a link is a fibre surface. This gives a short topological proof of the fact that a Murasugi is fibred if and only if its two summands are.