The Alexander polynomial

Graduation date: 2013The Alexander polynomial is a well understood classical knot invariant with interesting symmetry properties and recent applications in knot Floer homology. There are many different ways to compute the Alexander polynomial, some involving algebraic techniques and others more geometric or combinatorial approaches. This is an interesting example of how different types of mathematics can be used to describe the same result. While experts understand the relationships between different fields and methods of computation, the subtleties are often omitted in the literature. This paper describes four routes to the Alexander polynomial with the intent to explicate these subtleties and bring clarity to this intersection of subjects