Braids and configuration spaces

Abstract: A configuration space is a space whose points represent the possible states of a given physical system. As such they appear naturally both in theoretical physics and technical applications. For an example of the former, in analytical mechanics, the Lagrangian and Hamiltonian formulations of classical mechanics depend heavily on the use of a physical system’s configuration space for the description of its kinematical and dynamical behavior, and importantly, its evolution in time. As an example of a technical application, consider robotics, where the space of possible configurations of the mechanical linkages that make up a robot is an important tool in motion planning. In this case it is of particular interest to study the singularities of these mechanical linkages, to see if a given configuration is singular or not. This can be done with the help of configuration spaces and their topological properties. Arguably, the simplest configuration space possible arises when the system is just a collection of point-like particles in a plane. Despite its simplicity, the corresponding configuration space has substantial complexity and is of great interest in mathematics, physics and technology: For instance, it arises naturally in the mathematical modelling of robots performing tasks in a warehouse. In this thesis we go through the mathematics necessary to study the behaviour of paths in this space, which corresponds to motions of the particles. We use the theory of groups, algebraic topology, and manifolds to examine the properties of the configuration space of point-like particles in a plane. An important role in the discussion will be played by braids, which are certain collections of curves, interlaced in three-space. They are connected to many different topics in algebra, geometry, and mathematical physics, such as representation theory, the Yang-Baxter equation and knot theory. They are also important in their own right. Here we focus on their relation to configurations of points.