For plane curve singularities, the paper characterizes which arcs map to vanishing cycles under the geometric variation operator, and constructs 'topological exceptional collections' of vanishing arcsets for any A'Campo divide.
T\^ete-\`a-t\^ete graphs and twists
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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.
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Vanishing arcs for isolated plane curve singularities
For plane curve singularities, the paper characterizes which arcs map to vanishing cycles under the geometric variation operator, and constructs 'topological exceptional collections' of vanishing arcsets for any A'Campo divide.