On parametrized topological complexity

Motion planning problems aim to generate collision-free and safe paths for a robot between its initial and goal configurations. However, real environments are often not static; external conditions can change over time. For example, a submarine fleet navigating through mined waters must follow a safe route, even if the positions of obstacles change daily. In this thesis, we investigate the concept of parameterized topological complexity, which quantifies the complexity of the path generation process in such dynamic environments. First, the classical concept of topological complexity is discussed, followed by the introduction of its parameterized analog. The relationship between parameterized topological complexity and the notion of homotopic distance is elaborated. In this context, higher-dimensional generalizations of homotopic distance, as well as the notions of relative and higher-order topological complexity, are also examined. Finally, it is shown how parameterized topological complexity can be expressed in terms of homotopic distance. This thesis provides a basic introduction to the concept of parameterized topological complexity and demonstrates how it can be defined within the framework of homotopic distance.