Some tools for studying big state spaces

In motion planning, one often encounters high-dimensional state spaces that describe the configuration of things, be it the state of a robot(s), the planning necessary to autonomously drive a car or fly a drone. Many times these state spaces are difficult to come to terms with. There are however some powerful tools from topology for dealing very effectively with certain state-space “primitives”. These are some infinite-dimensional state spaces relevant to living in a 3-dimensional world. I will describe some tools and methods for navigating through the space of all ways one can knot certain types of rope.