Tête-à-tête : graphs and twists

This is a PhD thesis in the mathematical field of low-dimensional topology.
Its main purpose is to examine so-called tête-à-tête twists, which were defined by A'Campo. Tête-à-tête twists 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, tête-à-tête twists are twists around a graph.
It is shown that tête-à-tête 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.
We also see for some tête-à-tête twists how they can be used to generate the mapping class group of closed surfaces.
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 sum is fibred if and only if its two summands are.