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1996 | Buch

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Knot Theory and Its Applications

verfasst von: Kunio Murasugi

Verlag: Birkhäuser Boston

Buchreihe : Progress in Mathematics

Enthalten in: Professional Book Archive

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Über dieses Buch

Knot theory is a concept in algebraic topology that has found applications to a variety of mathematical problems as well as to problems in computer science, biological and medical research, and mathematical physics. This book is directed to a broad audience of researchers, beginning graduate students, and senior undergraduate students in these fields.

The book contains most of the fundamental classical facts about the theory, such as knot diagrams, braid representations, Seifert surfaces, tangles, and Alexander polynomials; also included are key newer developments and special topics such as chord diagrams and covering spaces. The work introduces the fascinating study of knots and provides insight into applications to such studies as DNA research and graph theory. In addition, each chapter includes a supplement that consists of interesting historical as well as mathematical comments.

The author clearly outlines what is known and what is not known about knots. He has been careful to avoid advanced mathematical terminology or intricate techniques in algebraic topology or group theory. There are numerous diagrams and exercises relating the material. The study of Jones polynomials and the Vassiliev invariants are closely examined.

"The book ...develops knot theory from an intuitive geometric-combinatorial point of view, avoiding completely more advanced concepts and techniques from algebraic topology...Thus the emphasis is on a lucid and intuitive exposition accessible to a broader audience... The book, written in a stimulating and original style, will serve as a first approach to this interesting field for readers with various backgrounds in mathematics, physics, etc. It is the first text developing recent topics as the Jones polynomial and Vassiliev invariants on a level accessible also for non-specialists in the field." -Zentralblatt Math

Inhaltsverzeichnis

Frontmatter

Introduction

Abstract

With a reasonably long, say 30cm in length, piece of string or cord, loosely bind a box as shown in Figure 0.1(a). You should now be holding in your hands a simple type of knot. Now take the two ends and glue them together so that it is not immediately noticeable that the string/cord has been joined. This exercise should be performed in such a way that the string does not come into contact with the box. The box is more a prop than a necessity. When the exercise is completed, what you should see before you is a single knotted loop, approximately 30cm long, Figure 0.1(b). In mathematics this loop is called a knot.

Kunio Murasugi

1. Fundamental Concepts of Knot Theory

Abstract

A knot, succinctly, is an entwined circle. However, throughout this book we shall think of a knot as an entwined polygon in 3-dimensional space, Figure 1.0.1(a). The reason for this is that it allows us, with recourse to combinatorial topology,¹ to exclude wild knots. For an example of a wild knot, consider the knot in Figure 1.0.1(b). Close to the point P, in a sense we may take this to be a “limit” point, the knot starts to cluster together in a concertina fashion. Therefore, in the vicinity of such a point particular care needs to be taken with the nature of the knot. We shall not in this exposition apply or work within the constraints of such (wild) knots. In fact, since wild knots are not that common, this will be the only reference to these kind of knots.

Kunio Murasugi

2. Knot Tables

Abstract

Knot theory, in essence, began from the necessity to construct knot tables. Towards the end of the 19^th century, several mathematical tables of knots were published independently by Little and Tait in British science journals. They managed to compile tables that in total consisted of around 800 knots, arranged in order from the simplest to the most “complicated.” However, since these tables included, for example, the two knots in Figure 1.2.1 as “distinct” knots, these tables were subsequently found to be incomplete. However, considering that these lists were compiled around 100 years ago, they are accurate to a very high degree. In this chapter we shall explain two typical methods of compiling knot tables.

Kunio Murasugi

3. Fundamental Problems of Knot Theory

Abstract

The problems that arise when we study the theory of knots can essentially be divided into two types. On the one hand, there are those that we shall call Global problems, while, in contrast, there are those that we shall call Local problems.

Kunio Murasugi

4. Classical Knot Invariants

Abstract

A knot (or link) invariant, by its very definition, as discussed in the previous chapter, does not change its value if we apply one of the elementary knot moves. As we have already seen, it is often useful to project the knot onto the plane, and then study the knot via its regular diagram. If we wish to pursue this line of thought, we must now ask ourselves what happens to, what is the effect on, the regular diagram if we perform a single elementary knot move on it? This question was studied by K. Reidemeister in the 1920s. In the course of time, many knot invariants were defined from Reidemeister’s seminal work. In this chapter, in addition to discussing these types of knot invariants, we shall also look at knot invariants that follow naturally from what one might say is mathematical experience.

Kunio Murasugi

5. Seifert Matrices

Abstract

In any science, in any discipline there are moments that can be called turning points — they reinvigorate and deepen the understanding of the subject at hand. What exactly is a turning point, even among friends, is usually contested and debated feverishly. Knot theory also has many turning points; however, there are two that are beyond debate: the Alexander polynomial and the Jones polynomial.

Kunio Murasugi

6. Invariants from the Seifert Matrix

Abstract

In order to find a knot (or link) invariant from a Seifert matrix, we need to look for something that will not change under the operations Λ₁ and $\Lambda^{\pm 1}_2$, defined in Theorem 5.4.1. We will see in this chapter that the Alexander polynomial is such an invariant. The Alexander polynomial is not the only important invariant that we can extricate from the Seifert matrix, the signature of a link can also be defined from it. In addition to defining these two invariants we shall, in this chapter, prove some of their basic characteristics. Nota bene, throughout this chapter we shall assume all the knots and links are oriented.

Kunio Murasugi

7. Torus Knots

Abstract

If we take two knots (or links) at random, what we would like to have is an efficient method that will determine for us whether or not they are equivalent knots (or links). In general, sadly, such an efficient method has yet to be discovered. So, at present a concise classification of knots is not possible. The next most obvious step is to try to group together knots (or links) with a particular property or properties in common, and then try to classify them. In fact, the techniques we have already discussed are sufficient for us to extract the characteristics of certain particular types of knots.

Kunio Murasugi

8. Creating Manifolds from Knots

Abstract

So far in this book we have concerned ourselves with the problem of classifying knots (and, of course, links). Intrinsically, this is a knot theoretical problem. This book, however, is twofold in nature and we wish to balance the purely theoretical with some practical applications of knot theory. The various applications of knot theory are discussed in detail in the latter chapters of this book; we would, however, in this chapter like to consider what might be called the classic application of knot theory. One of the most important, even fundamental, problems in algebraic topology is the general classification of manifolds (see Definition 8.0.1 below). In this chapter we will show that it is possible to create from an arbitrary knot (or link) a 3-dimensional manifold (usually shortened to 3-manifold). Hence by studying the properties of knots we can gain insight into the properties of 3-manifolds.

Kunio Murasugi

9. Tangles and 2-Bridge Knots

Abstract

During the period from the end of the 1960s through to the beginning of the 1970s, Conway pursued the objective of forming a complete table of knots. As we have seen in our discussions thus far, the knot invariants that had been discovered up to that point in time were not sufficient to accomplish this aim. Therefore, Conway pulled another jewel from his bag of cornucopia and introduced the concept of a tangle. Using this variation on a knot, a new class of knots could be defined: algebraic knots. By studying this class of knots, various Local problems were able to be solved, which led to a further jump in the level of understanding of knot theory. However, since there are knots that are not algebraic, the complete classification of knots could not be realized. Nevertheless, the introduction of this new research approach has had a significant impact on knot theory. In this chapter we shall investigate 2-bridge knots (or links), which are a special kind of algebraic knot obtained from trivial tangles.

Kunio Murasugi

10. The Theory of Braids

Abstract

At the beginning of the 1930s, as a means of studying knots, E. Artin introduced a concept of a (mathematical) braid(s). This remarkable insight itself was not sufficient to sustain research in this area, and so it slowly began to wither. However, in the 1950s this concept of braids was found to have applications in other fields, and this gave fresh impetus to the study of braids, rekindling research in this area. The iridescent hue of this concept flowering into full bloom and activity occurred in 1984, when V. Jones put into action with inordinate success the original aim of Artin, i.e., the application of braids to knot theory. In this chapter our intention is to introduce certain necessary aspects of the theory of braids that will prove useful when we explain recent developments in knot theory in the subsequent chapters.

Kunio Murasugi

11. The Jones Revolution

Abstract

In 1984, after nearly half a century in which the main focus in knot theory was the knot invariants derived from the Seifert matrix, for example, the Alexander polynomial, the signature of a knot, et cetera, V. Jones announced the discovery of a new invariant. Instead of further propagating pure theory in knot theory, this new invariant and its subsequent offshoots unlocked connections to various applicable disciplines, some of which we will discuss in the subsequent chapters.

Kunio Murasugi

12. Knots via Statistical Mechanics

Abstract

The motivation behind statistical mechanics is to try to understand, by using statistical methods, macroscopic properties — the easiest example being to determine what happens to water in a kettle when we boil it — by looking at the microscopic properties, i.e., how the various molecules interact. Statistical mechanics together with quantum mechanics have formed a basis for studying the physics of matter, i.e., the study from the atomic point of view of the various properties of matter. In general, the constituent molecules, even if we assume they obey the principles of dynamics, have extremely complicated means of motion. At present, mathematically these motions are virtually impossible to categorize. So, one reasonably successful method around this problem has been to form an ideal realization of matter. This realization takes the form of a statistical mechanical model that is a simplified copy of matter. The pivot that is essential for the model to at least have mathematical meaning is a function Z called the partition function,

$${\text{z = }}\sum\limits_\sigma {\exp \left( {\frac{{ - {\text{E}}\left( \sigma \right)}} {{k{\text{T}}}}} \right)},$$

in which we define σ to be a state of the particular model, E(σ) to be the total energy of this state, T to be the absolute temperature, and k to be Boltzmann’s constant. The sum itself is taken over all the states of the particular model.

Kunio Murasugi

13. Knot Theory in Molecular Biology

Abstract

F.H.C. Crick and J.D. Watson, in one of the most remarkable insights of the 20^th century, unraveled the basic structure of DNA. For this profundity into the substance of living matter, they were jointly awarded the Nobel Prize for Medicine in 1962. Essentially, a molecule of DNA may be thought of as two linear strands intertwined in the form of a double helix with a linear axis. A molecule of DNA may also take the form of a ring, and so it can become tangled or knotted. Further, a piece of DNA can break temporarily. While in this broken state the structure of the DNA may undergo a physical change, and finally the DNA will recombine. In fact, in the early 1970s it was discovered that a single enzyme called a (DNA) topoisomerase can facilitate this complete process, from the initial break to the recombination. The reader who might have picked up this book, looked at the title, and then randomly opened the book at this page may think that the publisher has somehow inserted some pages of an elementary textbook on biology here by mistake. But, let us reconsider the above. The double-helix structure of DNA — on some occasions DNA may even have only a single strand — is a geometrical entity, or more precisely, a topological configuration. This topological configuration is itself a manifestation of linking or knotting. Further, it has been shown when a topoisomerase causes DNA to change its form that the process is very similar to what happens locally in the skein diagrams.

Kunio Murasugi

14. Graph Theory Applied to Chemistry

Abstract

In our discussions thus far we have considered a graph to be a figure, to put it naively, composed of dots and line segments (topologically this is called a 1-complex). To be more exact, less intuitive, and more mathematical, a graph is usually thought of in an abstract sense. Therefore, strictly speaking, a (finite) graph G is a pair of (finite) sets {V_G, E_G} that fulfills an incidence relation. An element of V_G is then said to be a vertex of G, while an element of E_G is said to be an edge of G. The relation/condition mentioned above stipulates that an element, e, of E_G is incident to elements, say, a and b, of V_G (nota bene, the condition does not require a and b to be distinct.) The two vertices a and b are said to be endpoints of e. If it is the case that a = b, then e is said to be a loop.

Kunio Murasugi

15. Vassiliev Invariants

Abstract

Towards the end of the 1980s in the midst of the Jones revolution, V.A. Vassiliev introduced a new concept that has had profound significance in the immediate aftermath of the Jones revolution in knot theory [V]. The importance of these so-called Vassiliev invariants lies in that they may be used to study Jones-type invariants more systematically.

Kunio Murasugi

Backmatter

Titel: Knot Theory and Its Applications
verfasst von: Kunio Murasugi
Verlag: Birkhäuser Boston
Electronic ISBN: 978-0-8176-4719-3
Print ISBN: 978-0-8176-4718-6
DOI: https://doi.org/10.1007/978-0-8176-4719-3

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

Introduction

1. Fundamental Concepts of Knot Theory

2. Knot Tables

3. Fundamental Problems of Knot Theory

4. Classical Knot Invariants

5. Seifert Matrices

6. Invariants from the Seifert Matrix

7. Torus Knots

8. Creating Manifolds from Knots

9. Tangles and 2-Bridge Knots

10. The Theory of Braids

11. The Jones Revolution

12. Knots via Statistical Mechanics

13. Knot Theory in Molecular Biology

14. Graph Theory Applied to Chemistry

15. Vassiliev Invariants

Backmatter