An Introduction to Knot Theory

The mathematical theory of knots is intended to be a precise investigation into the way that 1-dimensional “string” can lie in ordinary 3-dimensional space. A glance at the diagrams on the pages that follow indicates the sort of complication that is envisaged. Because the theory is intended to correspond to reality, it is important that initial definitions, whilst being precise, exclude unwanted pathology both in the things being studied and in the properties they might have. On the other hand, obsessive concentration on basic geometric technology can deter progress. It can initially be but tasted if it seem onerous. At its foundations, knot theory will here be considered as a branch of topology. It is, at least initially, not a very sophisticated application of topology, but it benefits from topological language and provides some very accessible illustrations of the use of the fundamental group and of homology groups.

It will now be shown that any link in S ³ can be regarded as the boundary of some surface embedded in S ³. Such surfaces can be used to study the link in different ways. Here they are used to show that knots can be factorised into a sum of prime knots. Later they will feature in the theory and calculation of the Alexander polynomial.

The theory of the polynomial invented by V. F. R. Jones gives a way of associating to every knot and link a Laurent polynomial with integer coefficients (that is, a finite polynomial expression that can include negative as well as positive powers of the indeterminate). The association of polynomial to link will be made by using a link diagram. The whole theory rests upon the fact that if the diagram is changed by a Reidemeister move, the polynomial stays the same. The polynomial for the link is then defined independently of the choice of diagram. Thus, if two links can be shown, by means of specific calculation from diagrams, to have distinct polynomials, then they are indeed distinct links. This is a relatively easy way of distinguishing knots with diagrams of few crossings. Table 3.1 displays the Jones polynomials for the knots of at most eight crossings shown in Chapter 1. Those polynomials are, by easy inspection, all distinct, so the corresponding knots are all distinct. As will be observed, the Jones polynomial is good, but not infallible, at distinguishing knots. However, that is not its most exciting achievement. Other invariants have, particularly with the aid of computers, always managed to distinguish any interesting pair of knots. Some of those invariants will be encountered in later chapters. The Jones polynomial, however, has been used to prove pleasing new results concerning the possible diagrams that certain knots can possess (see Chapter 5). In addition, the Jones polynomial has been much generalised; it has been developed into a theory, allied in some sense to quantum theory, giving invariants for 3-dimensional manifolds (see Chapter 13) and has been the genesis of a remarkable resurgence of interest in knot theory in all its forms. It is amazing that so simple, powerful and provocative a theory remained unknown until 1984, [53]. Because of the ease with which it can be developed, understood and used, the Jones polynomial has a place very near to the beginning of any exposition of knot theory. The simplest way to define it is by using a slightly different polynomial: the bracket polynomial discovered by L. H. Kauffman [59].

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).

This chapter contains some of the most impressive applications of the Jones polynomial. They give solutions to two problems encountered by P. G. Tait in the nineteenth century. It is shown that an alternating knot diagram, when “reduced” in a rather elementary way, has the minimal number of crossings and that its writhe is an invariant of the knot. The crossing number of some other types of knot is also determined.

In order to bring to a satisfactory conclusion the theory of the last chapter, it is necessary to show that the space X _∞, together with the given action on it by the infinite cyclic group, is uniquely defined by the oriented link L under consideration. Here it will be seen that X _∞ is a certain covering space of the exterior of L, and the theory of coverings will show it to be well defined. That is the present motivation, but it should be understood that the theory of covering spaces is an important part of many areas of mathematics (particularly Riemann surfaces and geometric structures on manifolds). It is intimately related to the study of the (appropriately named) fundamental group of a fairly general type of topological space. Thus the following discussion will be in the language of general topological spaces.

The Conway polynomial [20] for an oriented link is really just the Alexander polynomial without the ambiguity concerning multiplication by units of ℤ[t ^-1, t]. Although that might seem a small improvement, it enables two such polynomials to be added together, which would be meaningless if the signs were in doubt, and this in turn permits a “skein formula” for the Alexander polynomials of links to be produced. The method for this given below uses Seifert matrices as before, but it abandons any interpretation by means of the homology of the infinite cyclic cover. (Use of the L-matrix of Reidemeister, as in [107] or [108], can also produce this theory.)

Most of this chapter will be concerned with a study of the twofold cyclic cover X ₂ → S ³ branched over an n component link L. The link L does not need to be oriented for this to make sense, but it will be sometimes convenient to select an arbitrary orientation in order to consider a Seifert surface. The principle result here is that the order of the first homology group H ₁ (X ₂) is det L—the determinant of the link, where det L = |Δ _L (−1)|—and that this number is often easy to calculate. As will be explained, the link determinant is, up to sign, the determinant of any Goeritz matrix [34] of the link, a matrix which is easy to write down starting from any diagram of the link.

The original Arf invariant was an invariant of certain quadratic forms on a vector space over a field of characteristic 2. This can be applied to a quadratic form, closely associated to the Seifert form, on the first homology with ℤ/2ℤ coefficients of a Seifert surface of an oriented link L. The result is a fairly classical link invariant A(L) ∈ ℤ/2ℤ called the Arf (or Robertello) invariant of L ([111], [114]). It must, however, be stated at once that for this theory to work—that is, for A(L) to be defined—L must satisfy the condition that the linking number of any component with the remainder of the link should be an even number. Before the discovery of the Jones polynomial, efforts to find a sensible generalisation of the Arf invariant to all links met with no success. The Jones polynomial V (L) is, of course, always defined. As will be shown in what follows, evaluating V(L) when t = i (with t ^1/2 = e ^iπ/4)gives where #L is the number of components of L. In a sense, this shows why a definition of A(L) for any link could not be found. Interpreted from the point of view of the Jones polynomial, this result gives one of the very few evaluations of the polynomial in terms of previously known invariants that can be calculated in “polynomial time” (see Chapter 16). This chapter will first explore the Arf invariant for vector spaces over ℤ/2ℤ and then effect liaison with the Jones polynomial.

It is in its interaction with the theory of the fundamental group that the theory of knots and links becomes almost a part of the general theory of 3-manifolds. It is the exterior of a link (that is, the closure of the complement in S ³ of a small regular neighbourhood of the link) that is studied, by means of its group, as a compact 3-manifold with torus boundary components. In the theory of 3-manifolds this is a very important example, but perhaps not much more than that. Here the view has been taken that to a mathematician it is the proving of results that brings satisfaction, and that this is particularly important in knot theory, wherein a cheerful punter might be satisfied by a good diagram. However, 3-manifold theory is well documented at length elsewhere ([43], [49]), and other more established treatises on knots have dwelt comprehensively on the relationship between links and the fundamental group. Thus what follows in this chapter is but an essay on this topic. It tries to interpret the Alexander polynomial in terms of the fundamental group and to explain what is available in more detail elsewhere.

The aim of this chapter is to show, in Theorem 12.14, that every closed connected orientable 3-manifold can be obtained by “surgery” on S ³. The method used is a version of that of [77]. An elementary r-surgery on a general n-manifold M is the operation of removing from M an embedded copy of S ^r × D ^n-r and replacing it with a copy of D ^r+1 × S ^n-r-1 the replacement being effected by means of the obvious homeomorphism between the boundaries of the removed set and its replacement. Surgery in general is a sequence of elementary surgeries. In the case of surfaces, instances of 1-surgery and 0-surgery have already been employed in earlier chapters, usually when the surface was contained in S ³. The only surgeries needed in this chapter are 1-surgeries on a 3-manifold, and it is easy to see they can be performed “simultaneously”. The surgery process will consist of the removal from S ³ of disjoint copies of S ¹ × D ² and their replacement by copies of D ² × S ¹ Of course, the set removed and its replacement are homeomorphic, but the parametrisation of the removed set as disjoint copies of S ¹ × D ², and the canonical method of replacement with respect to that, ensure that the new manifold is usually not S ³. A collection of disjoint solid tori in S ³ is just a regular neighbourhood of a link, and a parametrisation of a neighbourhood of each component by S ¹ × D ² is called a framing of the link. Thus it will be shown that 3-manifolds can be interpreted by means of framed links in S ³.

As proved in Chapter 12, any closed connected orientable 3-manifold can be obtained by the process of surgery on a framed link in S ³. Any invariant of framed links can be applied to such a surgery prescription in the hope of finding an invariant of the 3-manifold. That would need to be some entity associated to the 3-manifold and not just to the particular surgery description; it would need to be unchanged by all possible Kirby moves. An elementary example comes from the idea of linking numbers. A framed link (with components temporarily ordered and oriented) has a linking matrix. This is the symmetric matrix with entries the linking numbers between the pairs of components of the link. The linking number of a component with itself (a diagonal term of the matrix) is taken to be the integer that gives the framing of that component. This linking matrix can easily be seen to be a presentation matrix (in the sense of Chapter 6) for the first homology of the 3-manifold arising from surgery on the framed link. Thus the modulus of the determinant of the matrix, if it is non-zero, is the order of that homology group and the nullity of the matrix is the first Betti number of the manifold. It is easy to check that these numerical invariants do indeed remain unchanged by Kirby moves on the framed link. This, however, is not too exciting, as homology is long and better understood by other means. One might hope to emulate this procedure by a simple direct application of some link invariant. The Alexander polynomial and the Jones polynomial fail in that respect. This chapter explains how the Jones polynomial can nevertheless be amplified to achieve a 3-manifold invariant. Roughly, the idea is to take a linear sum of the Jones polynomials, evaluated at a complex root of unity, of copies of the link with the components replaced by various parallels of the original components. The resulting invariants are known as Witten’s quantum SU _q (2) 3-manifold invariants. The details are somewhat intricate and, as might be expected, will here be eased by the simplifying approach of the Kauffman bracket and the linear skein theory associated with it. The Temperley-Lieb algebras appear as instances of that theory. E. Witten’s initiation of this topic can be found in [135].

The quantum SU _q (2) 3-manifold invariants associated with a primitive 4r ^th root of unity, described in the previous chapter, are fairly new and mysterious. Their use has so far been exceedingly limited in knot theory and in 3-manifold theory. Certainly they do distinguish many pairs of 3-manifolds, even pairs with the same homotopy type, but that has usually been more simply achieved by other means. However, there exist pairs of distinct manifolds with the same invariants for all r (see [85], [55] and [62]). For some manifolds, for some values of r the invariant is known by direct calculation to be zero. Superficially it might seem to be almost impossible to calculate any of these invariants. The calculation, from first principles, of the invariant corresponding to a 4r ^th root of unity involves taking an (r-2)-parallel of a surgery link giving the 3-manifold. If the link’s diagram has n crossings, that of the parallel has n(r - 2)² crossings; calculating a Jones polynomial by naive means soon becomes impractical when many crossings are involved. It will be shown here that it is in principle fairly easy to give a formula, as a summation, for the invariants of lens spaces and, more generally, for certain Seifert fibrations. Although in theory any of the invariants can always be calculated, it is sensible to use various simplifying procedures whenever possible. Some of those will be described in this chapter. Tables of specific computer calculations appear in [104] and in [62], where one can search for patterns in the resulting lists of complex numbers.

Elementary properties of the Jones polynomial have already been discussed in Chapter 3. Versions of some of those results which hold equally well for the HOMFLY and Kauffman polynomials are given below. Where proofs are essentially the same as those relating to the Jones polynomial, they are left as an exercise.