Contact open books: Classical invariants and the binding sum

In the light of the Giroux correspondence of open books and contact structures, it is natural to study Legendrian knots embedded in a page of a compatible open book of a contact 3-manifold. Legendrian knots, as well as their classical invariants, provide useful information on the ambient contact manifold. This thesis develops formulas for deciding whether a given knot lying on a page of a compatible open book whose monodromy is encoded as a concatenation of Dehn twists along non-isolating curves is nullhomologous, and, if so, for computing its classical invariants. Similarly, we show how to compute the Poincaré dual of the Euler class of the contact structure and how to compute the d3-invariant of the contact structure in case the Euler class is torsion. All invariants can directly be computed from data included in the open book, namely via the intersection behaviour of the knot, an arc basis of the page and the Dehn twist curves encoding the monodromy. We then turn to higher-dimensional manifolds, for which the relation of open books and contact structures remains partly intact. In particular, every contact structure on a manifold is supported by a compatible open book. First purely topologically and also in even dimensions, we study fibre connected sums in the context of open book decompositions and introduce a new class of submanifolds, called nested open books, which are particularly well adapted to this setting. We show that the fibre connected sum of an open book along diffeomorphic binding components -- called the binding sum -- admits a natural open book decomposition provided the respective binding components admit open book structures themselves, which is no restriction in odd dimensions. Furthermore, we prove that in case the binding sum is performed in a contact open book supporting a given contact structure on the manifold, the construction can be adapted such that the resulting natural open book decomposition is compatible with the contact structure obtained by the usual contact fibre connected sum along contactomorphic binding components