A Survey of Knot Theory

Frontmatter

Chapter 0. Fundamentals of knot theory

Abstract

In this chapter, we first explain the PL category in which we consider spaces. Next, PL manifolds and related matters are defined. Finally, PL knots and PL links are defined together with related basic concepts.

Akio Kawauchi

Chapter 1. Presentations

Abstract

In this chapter, we discuss regular presentations, braid presentations and bridge presentations for links.

Akio Kawauchi

Chapter 2. Standard examples

Abstract

In this chapter, we discuss 2-bridge links, torus links and pretzel links. These links appear very often in studies on knot theory.

Akio Kawauchi

Chapter 3. Compositions and decompositions

Abstract

In this chapter, we discuss how to construct a new link from given links by various compositions. Then we discuss decompositions, which are the inverse operations of compositions. After that, compositions of tangles are discussed. Throughout this chapter, links are understood to be links in S³.

Akio Kawauchi

Chapter 4. Seifert surfaces I: a topological approach

Abstract

In this chapter, Seifert surfaces for links are introduced. The notions of an incompressible Seifert surface, a minimal genus Seifert surface, and a fiber Seifert surface are discussed with respect to Murasugi sums.

Akio Kawauchi

Chapter 5. Seifert surfaces II: an algebraic approach

Abstract

In this chapter, we discuss the Seifert matrix, which is derived from a connected Seifert surface of a link, and related link invariants such as the signature, the nullity, the Arf invariant and the one-variable Alexander polynomial.

Akio Kawauchi

Chapter 6. The fundamental group

Abstract

In this chapter, we discuss various properties of the fundamental group of a link exterior.

Akio Kawauchi

Chapter 7. Multi-variable Alexander polynomials

Abstract

In this chapter, we define the Alexander module and the link module of a link and show how to calculate them by Fox’s free differential calculus. Then we define the (multi-variable) graded Alexander polynomials to be the characteristic polynomials of these modules and explain the Torres conditions, which the (0-th) Alexander polynomial satisfies.

Akio Kawauchi

Chapter 8. Jones type polynomials I: a topological approach

Abstract

In this chapter, we discuss the following polynomial invariants of a link: the Conway polynomial, the Jones polynomial, the skein polynomial, the Q polynomial and the Kauffman polynomial.

Akio Kawauchi

Chapter 9. Jones type polynomials II: an algebraic approach

Abstract

In this chapter, we discuss some algebras related to link polynomials such as the skein polynomial in order to explain how the polynomials arise from the representation theory of algebras.

Akio Kawauchi

Chapter 10. Symmetries

Abstract

As shown in figure 10.0.1, there are various kinds of symmetries on knots. In the first half of this chapter, we study some relationships between symmetries and the polynomial invariants. As an application, we explain the proof of [Kawauchi 1979] on the non-invertibility of 8₁₇ (see figure 10.0.2). In the latter half of this chapter, we study the symmetry group of a knot, which essentially controls the symmetries of a knot. We explain a (still unpublished) theory of F. Bonahon and L. Siebenmann (cf. [Bonahon-Siebenmann *]) for a canonical decomposition of a knot, which gives us good insight into the knot and enables us to determine the symmetry groups of algebraic knots including 817 and the Kinoshita-Terasaka knot K _KT (see figure 3.8.1a).

Akio Kawauchi

Chapter 11. Local transformations

Abstract

In this chapter, we discuss several patterns of local transformations on link diagrams, the major theme of which is unknotting operations on knots.

Akio Kawauchi

Chapter 12. Cobordisms

Abstract

In this chapter we discuss the concept of knot cobordism, which is a 4-dimensional property of a knot. In the latter half, this concept is generalized to links.

Akio Kawauchi

Chapter 13. Two-knots I: a topological approach

Abstract

In this chapter, we discuss a normal form for 2-knots, how to construct 2-knots, and some properties of ribbon 2-knots. Most results are described without proof, but the reader can consult the papers and books that are cited there or in the supplementary notes of this chapter.

Akio Kawauchi

Chapter 14. Two-knots II: an algebraic approach

Abstract

By an n-knot group, we mean the fundamental group π₁ (S ⁿ⁺² - K ⁿ , b) for an n-knot K ⁿ in S ⁿ⁺² . Similarly, by a surface-knot group, we mean the fundamental group π₁ (S ⁴ - F, b) for an oriented surface-knot F in S ⁴ . In this chapter, we discuss some properties of 2-knot groups in comparison with those of the other dimensional knot groups and surface-knot groups.

Akio Kawauchi

Chapter 15. Knot theory of spatial graphs

Abstract

The topological study of spatial graphs is considered to be a natural extension of knot theory, although it has not been paid much attention until quite recently. In this chapter, we regard two notions on “equivalence” of graphs. The first one is a notion naturally extending positive-equivalence of links and is called equivalence. The second one is a notion which is useful when we study the exterior of a spatial graph and is called neighborhood-equivalence. Since the importance of the first concept is motivated by recent developments in molecular chemistry, we devote the first section to some comments on the topology of molecules. In 15.2 we discuss some results on the first notion, and in 15.3 some results on the second notion, including an explanation of recent developments on the tunnel number.

Akio Kawauchi

Chapter 16. Vassiliev-Gusarov invariants

Abstract

In this chapter, we discuss a graded Q-algebra of numerical link invariants which we call the Vassiliev-Gusarov invariants (cf. [Vassiliev 1990], [Gusarov 1994, 1994′]). An important observation is that this algebra determines the Jones, skein and Kauffman polynomials and their satellite version invariants, as is discussed in 16.2. In 16.3, we discuss Kontsevich’s iterated integral invariant, and characterize the Vassiliev-Gusarov algebra in terms of the weight systems on chord diagrams. In 16.4, we discuss numerical link invariants which are not of Vassiliev-Gusarov type.

Akio Kawauchi

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