Combinatorial Properties of Ladders, Generalised Pont-Neuf and Caterpillar Diagrams in the Space of Coloured Open Jacobi Diagrams

This doctoral thesis is a contribution to the analysis of the combinatorics of arbitrarily coloured open Jacobi diagrams and their relationship to Vassiliev invariants. We examine J.~Kneissler's five ladder relations and state them in a much more precise way. We also analyse their role in the space of colored open Jacobi diagrams. Then, we establish a sort of machinery - a language together with a toolbox of lemmata, theorems and definitions to build, manipulate and analyse coloured open Jacobi diagrams. With this, we examine the role of generalised Pont-Neuf diagrams and caterpillar diagrams. Lastly we transfer this to the uncolored case, which allows us to show that the space of open Jacobi diagrams up to first Betti number five is already contained in the module of caterpillar diagrams, considered as a module of a certain subset of Vogels' algebra. This means that Vassiliev invariants associated to these degrees do not detect knot orientation.