This project will focus on studying the fundamental groups of topological spaces. The goal is to specifically use ideas from group theory to differentiate between different types of knots by using the fundamental group as a topological invariant. First, we aim to provide a background in topology, including introducing deformation retractions and homotopy types. We will then explore new algebraic concepts, such as free groups and group presentations. This will allow us to develop a general understanding of how to find the fundamental group of a topological space and how to use it to gain more insight into which spaces are homeomorphic. A key theorem to finding the fundamental group of such spaces is the Seifert and Van Kampen Theorem. Next, we will apply the fundamental group to knots using various methods, such as the Wirtinger presentation. The fundamental group will provide information about the knots’ homotopy types and by developing a presentation of a knot, we will be able to distinguish between some different knot types.
Solomon, H. M. (2018). The Fundamental Group and Knots (Undergraduate honors thesis, University of Redlands). Retrieved from https://inspire.redlands.edu/cas_honors/176