Algebraic Invariants of Links

The following sections are included:

• Contents

• Preface

In this chapter we shall define knots and links and the standard equivalence relations used in classifying them. We shall also outline the most important geometric aspects. The later chapters shall concentrate largely on the algebraic invariants of covering spaces.

The primary algebraic invariants of knots and links are the homology groups of covering spaces of the exterior, considered as modules over the group ring of the covering group, together with the bilinear pairings determined by Poincaré duality. In high dimensions simple knots (i.e., knots with highly connected Seifert surfaces) are completely classified by such invariants, and the knot concordance group is isomorphic to an algebraically defined Witt group of equivalence classes of pairings.

The main determinantal invariants of modules and chain complexes over a noetherian ring are the elementary ideals, their divisorial hulls, Reidemeister-Franz torsion and the Steinitz-Fox-Smythe invariant. We shall describe these, and also consider some special features of low-dimensional rings and of Witt groups of hermitean pairings on torsion modules. Our principal technique shall be to reduce to the case of a discrete valuation ring by localizing at height 1 prime ideals. (In our applications the ring shall usually be a quotient of a ring of Laurent polynomials over ℤ or over a field.)

In this chapter we shall define the Crowell sequence for a group and extend to links various results of Crowell on the Alexander modules of classical knots. We shall focus particularly on the interactions between properties known to hold for boundary links, such as the Alexander module having rank μ, the first nonzero Alexander ideal being principal and the longitudes being in (π_ω)′.

The Torres conditions relate the Alexander invariants of a link to those of its sublinks. Similar reductions give invariants of intermediate abelian covering spaces. We shall consider in some detail the homology of abelian branched coverings of homology 3-spheres, branched over links.

The Alexander invariants of a 1-link L are invariants of π/π″, where π = πL. They are readily computable, in terms of determinants of matrices over a commutative ring. However, they are of little use if, for instance, π′ is perfect. Representations into finite groups are computable, but do not dig very deeply into the group. Taken in conjunction with a homomorphism onto a free abelian group they give a manageable class of representations which penetrate below the commutator group. Twisted Alexander invariants combine features of representability into well-understood non-solvable groups (such as linear groups) with computability of invariants defined in terms of matrices over a commutative ring.

In this chapter and the next we shall consider the special cases μ = 1 and μ = 2, respectively. The knot theoretic case has been the most studied. In higher dimensions knot modules and the associated pairing provide complete invariants for significant classes of knots ([Le70, Fa83]), and the Witt class of the Blanchfield pairing on the middle dimensional knot module is a complete invariant of concordance for (2q + 1)-dimensional knots with q > 1 [Le69, Ke75’]. The question of which modules and pairings are realized by highdimensional knots has been almost completely answered by Levine [Le77], and the algebraic study of such modules was pursued in [Lev]. We shall make some complementary observations on the structure of knot modules, and give some further detail on Blanchfield pairings and cyclic branched covers for classical knots. In the final section we use ribbon links to realize arbitrary Laurent polynomials with augmentation 1 as the first nonzero Alexander polynomials of the groups of higher-dimensional links.

Linking as distinct from knotting is first apparent when μ = 2. In this case there are also some simplifications and special results, both in the algebra and the topology. Bailey has characterized the Λ₂-modules which arise as B(L) for some 2-component link L as the modules having certain presentation matrices. Link-homotopy is determined by linking number alone. On the other hand, the classification of links up to I-equivalence is poorly understood, even in the 2-component case.

The word “symmetry” may be interpreted in several ways: we shall consider both questions dealing with symmetries of link types such as whether a link is invertible, amphicheiral or interchangeable, and also those dealing with group actions on a particular link. In §2 and §4 we sketch some results on amphicheirality and invertibility of simple (2k − 1)-knots. Most of the rest of this chapter is devoted to group actions on classical links. The basic definitions and consequences of the Smith Conjecture are given in §3. In §5, §6 and §7 we assume that the action is defined by rotation about a disjoint axis, and in §8 we consider free actions. In the final section we return to knots invariant under rotations.

In this chapter we shall give an account of plane curve singularities from the point of view of elementary commutative algebra. The geometric and topological aspects of plane curves and their singularities are treated in much greater detail in the books [BK], [EN] and [Mil].

From the homological perspective the significant feature of the group Z is that it is free, rather than that it is abelian. The Laurent polynomial ring Λ_μ = ℤ[ℤ^μ] is a commutative factorial noetherian domain, but it has global dimension μ + 1, and so homological arguments can become unwieldy for μ large. In contrast, while the group ring ℤ[F(μ)] of a finitely generated non-abelian free group is no longer commutative or noetherian, it is coherent and has global dimension 2, for any value of μ.

The theorem of Stallings (Theorem 1.3) demonstrates a close connection between the homology of a group and its lower central series. To what extent can we derive the nilpotent quotients homologically? The answer is that, roughly speaking, the Massey product structure on H¹(G;ℚ) determines the rational Mal’cev completion of G (defined below).

This chapter shall summarize work of Cochran, Le Dimet, Levine, Orr and Vogel on universal groups for links whose groups have relatively free nilpotent quotients ( for 1 ≤ k < ∞). This includes all 1-links I-equivalent to sublinks of homology boundary links and all higher dimensional links.

One serious difficulty in attempting to classify links up to concordance is that the set of concordance classes does not have a natural group structure. Le Dimet, working in higher dimensions, and subsequently Levine and Habegger and Lin, in the classical case, have shown that one should first study disc links (or “string links”). This chapter sketches some of their work.

The following sections are included:

• Bibliography

• Index