The Knot Group and The Jones Polynomial

In this thesis, basic knot theory is introduced, along with concepts from topology, algebra and algebraic topology, as they relate to knot theory. In the first chapter, basic definitions concerning knots are presented. In the second chapter, the fundamental group is applied as a method of distinguishing knots. In particular the torus knots are classified using the fundamental group, and a general algorithm for computing the group of certain well-behaved knots is shown to be successful. In the third chapter, the Jones polynomial is developed by considering diagrams of oriented links, and shown to be unaffected by certain changes in these diagrams. Finally, the relative strengths and weaknesses of the Jones polynomial and the knot group are discussed.