Tangles and 2-Bridge Knots

During the period from the end of the 1960s through to the beginning of the 1970s, Conway pursued the objective of forming a complete table of knots. As we have seen in our discussions thus far, the knot invariants that had been discovered up to that point in time were not sufficient to accomplish this aim. Therefore, Conway pulled another jewel from his bag of cornucopia and introduced the concept of a tangle. Using this variation on a knot, a new class of knots could be defined: algebraic knots. By studying this class of knots, various Local problems were able to be solved, which led to a further jump in the level of understanding of knot theory. However, since there are knots that are not algebraic, the complete classification of knots could not be realized. Nevertheless, the introduction of this new research approach has had a significant impact on knot theory. In this chapter we shall investigate 2-bridge knots (or links), which are a special kind of algebraic knot obtained from trivial tangles.