Date of Award
2018
Document Type
Thesis
Degree Name
Bachelors
Department
Natural Sciences
First Advisor
McDonald, Patrick
Area of Concentration
Mathematics
Abstract
This thesis is intended to explore the use of the Alexander polynomial as it applies to knot theory, and to build up the background in algebraic topology necessary to discuss some of the main concepts of this topic. After defining some topological concepts, we cover the fundamental group and singular homology groups, providing definitions, examples and computational methods. Next, the formal definition of knot is developed. We then introduce the knot group and give a method to calculate it. Finally, we construct the infinite cyclic covering of the knot complement and use it to define the Alexander polynomial. We then walk through an algorithm for computing the Alexander polynomial given a knot diagram, and use it to find the Alexander polynomial of three different knots.
Recommended Citation
Price, Jacob, "KNOT THEORY AND THE ALEXANDER POLYNOMIAL" (2018). Theses & ETDs. 5583.
https://digitalcommons.ncf.edu/theses_etds/5583