Geometry of Alternating Links

Abstract

An alternating diagram for a link is, as explained in Chapter 1, one in which the over or under nature of the crossings alternates along every link-component in the diagram; the crossings always go “… over, under, over, under,…” when considered from any starting point. A link is said to be alternating if it possesses such a diagram. It has long been realised that alternating diagrams for a knot or link are particularly agreeable. However, the question posed by R. H. Fox—“What is an alternating knot?”—by which he was asking for some topological characterisation of alternating knots without mention of diagrams, is still unanswered. In later chapters the way in which the alternating property interacts with polynomial invariants will be discussed. In what follows here, some of the geometric properties of alternating links, discovered by W. Menasco [94], will be considered. The results are paraphrased by saying that an alternating link is split if and only if it is obviously split and prime if and only if it is obviously prime. Here “obviously” means that the property can at once be observed in the alternating diagram. This then establishes a ready supply of prime knots. Much of the ensuing discussion will concern 2-spheres embedded in S ³. It is to be assumed, as usual, that all such embeddings are piecewise linear (that is, simplicial with respect to some subdivisions of the basic triangulations).