The Fundamental Group of a Link Notes for Math 148

Associated to every knot or link in R (or in S) we can associate a group, called the fundamental group of the complement of the link, or more simply, the fundamental group of the link. A simple-minded way to approach this is to extend the idea of labeling a link diagram by group elements. Given a group G and an oriented link diagram D, we say that D can be labeled with G if we can label every arc of the diagram with an element from G such that