The Geometry of Corkscrew Tangles

This project highlights the connections between elastic spaces, known as topological spaces, and spaces with rigid geometric structure. It is an investigation of an infinite subset of topological spaces not yet studied which are known as corkscrew tangles. These are formed by “drilling” (conceptually) two tunnels out of a solid ball, from which the geometric structure of the tangles may be deduced. Preliminary efforts have led to a description of the geometry of the simplest tangles in the family. Furthermore, a consistent method has been found to triangulate arbitrary tangles and this seems to be the key to finding geometric structures for the remaining tangles. Three computer programs assist with computation necessary for the project: SnapPy, Regina, and Sage. Snappy provides information such as the volume of the tangles. Regina gives detailed descriptions about the “triangulation” of the space. Sage, a computer algebra system, is used to solve large systems of equations. This project integrates computer skills, mathematical intuition and critical thinking skills and applies them to a contemporary issue in topology. The primary objective of the project is to develop a generalization of the geometry of these corkscrew tangles, which may be achieved by the time of the UMCUR conference