Knot Theory

Knot Theory

CHARLES LIVINGSTON
Volume: 24
Copyright Date: 1993
Edition: 1
Pages: 259
https://www.jstor.org/stable/10.4169/j.ctt5hh90k
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  • Book Info
    Knot Theory
    Book Description:

    Knot Theory, a lively exposition of the mathematics of knotting, will appeal to a diverse audience from the undergraduate seeking experience outside the traditional range of studies to mathematicians wanting a leisurely introduction to the subject. Graduate students beginning a program of advanced study will find a worthwhile overview, and the reader will need no training beyond linear algebra to understand the mathematics presented. The interplay between topology and algebra, known as algebraic topology, arises early in the book, when tools from linear algebra and from basic group theory are introduced to study the properties of knots. Livingston guides you through a general survey of the topic showing how to use the techniques of linear algebra to address some sophisticated problems, including one of mathematics' most beautiful topics, symmetry. The book closes with a discussion of high-dimensional knot theory and a presentation of some of the recent advances in the subject — the Conway, Jones, and Kauffman polynomials. A supplementary section presents the fundamental group, which is a centerpiece of algebraic topology.

    eISBN: 978-1-61444-023-9
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-viii)
  2. ACKNOWLEDGEMENTS
    (pp. ix-x)
  3. PREFACE
    (pp. xi-xiv)
  4. Table of Contents
    (pp. xv-xviii)
  5. CHAPTER 1: A CENTURY OF KNOT THEORY
    (pp. 1-10)

    In 1877 P. G. Tait published the first in a series of papers addressing the enumeration of knots. Lord Kelvin’s theory of the atom stated that chemical properties of elements were related to knotting that occurs between atoms, implying that insights into chemistry would be gained with an understanding of knots. This motivated Tait to begin to assemble a list of all knots that could be drawn with a small number of crossings. Initially the project focused on knots of 5 or 6 crossings, but by 1900 his work, along with that of C. N. Little, had almost completed the...

  6. CHAPTER 2: WHAT IS A KNOT?
    (pp. 11-28)

    There axe many definitions of knot, all of which capture the intuitive notion of a knotted loop of rope. For each definition there is a corresponding definition of deformation, or equivalence. This chapter will concentrate on one pair of such definitions, and mention another. (Results at the foundations of geometric topology relate the various definitions. Such matters will not be presented here, and do not affect the work that follows.) The goal for now is to demonstrate how the notion of knotting can be given a rigorous mathematical formulation, and to give the reader a flavor of the problems and...

  7. CHAPTER 3: COMBINATORIAL TECHNIQUES
    (pp. 29-54)

    The techniques of knot theory which are based on the study of knot diagrams are called combinatorial methods. These techniques are usually easy to describe and yet provide deep results. For instance, in this chapter such methods will be used to prove that nontrivial knots exist and then to demonstrate that there is in fact an infinite number of distinct knots.

    Combinatorial tools often appear as unnatural or ad hoc. In many cases alternative perspectives, though more abstract, can provide insights. One of the successes of algebraic topology is to provide such perspectives, but in some cases, the efficacy of...

  8. CHAPTER 4: GEOMETRIC TECHNIQUES
    (pp. 55-82)

    Consider the surface drawn in Figure 4.1. It is built from a disk by attaching two twisted bands. Note that the boundary, or edge, of the surface is a knotted curve. In fact, the boundary is a trefoil knot.

    By studying the surface it is possible to learn more about the trefoil knot. In general, the termgeometric techniquesrefers to the methods of knot theory that are based on working with surfaces. The use of these methods is motivated by a theorem stating that for every knot there is some surface having that knot as its boundary. An important...

  9. CHAPTER 5 ALGEBRAIC TECHNIQUES
    (pp. 83-108)

    The field of mathematics called algebraic topology is devoted to developing and exploring connections between topology and algebra. In knot theory, the most important connection results from a construction which assigns to each knot a group, called thefundamental group of the knot. Knot groups will be developed here using combinatorial methods. An overview of the general definition of the fundamental group is given in the final section of the chapter.

    The fundamental group of a nontrivial knot typically is extremely complicated. Fortunately, its properties can be revealed by mapping it onto simpler, finite, groups. The symmetric groups are among...

  10. CHAPTER 6: GEOMETRY, ALGEBRA, AND THE ALEXANDER POLYNOMIAL
    (pp. 109-128)

    The discovery of connections between the various techniques of knot theory is one of the recurring themes in this subject. These relationships can be surprising, and have led to many new insights and developments. A recent example of this occurred with the discovery by V. Jones of a new polynomial invariant of knots. Although his approach was algebraic, the Jones polynomial was soon reinterpreted combinatorially. Almost immediately there blossomed an array of new combinatorial knot invariants which appear to be among the most useful tools available for problems relating to the classification of knots. An understanding of these new invariants...

  11. CHAPTER 7: NUMERICAL INVARIANTS
    (pp. 129-150)

    A few methods for associating integers to knots have already appeared in the text. The genus is an important example. Others include the signature, the determinant, and the modprank. In this chapter many more will be described. Some of these will seem to be very natural quantities to study. Others, such as the degree of the Alexander polynomial, may at first seem artificial; it is the relationship between these invariants and the more natural ones that is particularly interesting and useful.

    It will be clear in this chapter that with the introduction of each new invariant a host...

  12. CHAPTER 8: SYMMETRIES OF KNOTS
    (pp. 151-178)

    Knot diagrams can appear symmetrical, and for those that do not, the lack of symmetry is often an artifact of the diagram, and is not inherent in the knot itself. For instance, Figure 8.1 presents two diagrams for the knot 7₆. The first shows no apparent symmetry, while the second is quite symmetrical; a rotation of 180 degrees about a point in the plane leaves the diagram unchanged. As the example indicates, finding symmetrical diagrams for a knot can be a challenging task. On the other hand, powerful tools are available for proving that a knot does not have hidden...

  13. CHAPTER 9: HIGH-DIMENSIONAL KNOT THEORY
    (pp. 179-204)

    The theory of knots inR³ naturally generalizes to a study of knotting inRn, withn> 3, and many new and fascinating aspects of knot theory appear in this high-dimensional setting. What is perhaps most surprising is that many problems that are intractable in the classical case have been solved for high-dimensional knots. There is also a strong interplay between knot theory in different dimensions, and this interplay leads to an array of new topics at the border of the classical and high-dimensional settings.

    The definitions of polygonal knot and of deformation of knots generalizes immediately toR⁴,...

  14. CHAPTER 10: NEW COMBINATORIAL TECHNIQUES
    (pp. 205-220)

    New combinatorial knot invariants have been discovered which are simple in definition and yet extremely powerful. Unlike those described earlier, there is no known connection to knot theory in higher dimensions. It now seems likely that they relate to properties that are unique to dimension 3. The new techniques have their roots in an observation made by Alexander in his original paper on the Alexander polynomial, an observation that went unexploited for forty years.

    Given an oriented link diagram, focus on a particular crossing. If that crossing is changed from right to left or vice versa, a new link diagram...

  15. APPENDIX 1: KNOT TABLE
    (pp. 221-228)
  16. APPENDIX 2 ALEXANDER POLYNOMIALS
    (pp. 229-232)
  17. REFERENCES
    (pp. 233-238)
  18. INDEX
    (pp. 239-240)