The proceeding of this conference was knots 90 by akio kawauchi, e. Kauffman, is a professor in the department of mathematics, statistics, and computer science at the university of illinois at chicago. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined together so that it cannot be undone, the simplest knot being a ring or unknot. A knot in r 3 or alternatively in the 3sphere, s 3, can be projected onto a plane r 2 respectively a sphere s 2. In this sense the mark is a formal model of a majorana fermion. The lewis theory used observations from chemists and physicists to form a theory about chemical bonding. In mathematics, we mean something slightly di erent by a knot. The author has provided a new supplement, entitled remarks on formal knot theory, as well as his article, new invariants in the theory of knots, first.
This paper is an introduction to the state sum model for the alexanderconway polynomial that was introduced in the the authors book formal knot theory princeton university press, 1983. Example applications of braid groups include knot theory, where any knot may be represented as the closure of certain braids a result. In the formalist point of view a specific mathematical game formal system can itself be an object of study for the mathematician. Reidemeister and alexander around about 1930 were able to make signi cant progress in knot theory. Also we will give a short introduction to knot theory and give a description of the most important knot invariants. Freediffcalc notes of fox calculus from the book on knots by lk. This means that we shall give a formula for the alexander polynomial that is a sum of evaluations of combinatorial con. The only formal prerequisite will be to have taken 868869 or the equivalent.
In the formalist point of view a specific mathematical game formal sys tem can itself be. The jones polynomial of an alternating link 41 exercises 48 chapter 6. By the early 1990s, knot theory was recognized as another such area of mathe. An introduction to knot theory university of newcastle.
Two knots are defined to be equivalent if there is an ambient isotopy between them projection. Seifert surfaces and knot factorisation 15 exercises 21 chapter 3. We have also avoided 4dimensional questions, such as the sliceribbon conjecture problem 1. He is the editorinchief of jktr, editor of the series on knots and everything, full professor at uic and author of numerous books related to the theory of knots including knots and physics, knots and applications, on knots, formal knot theory. Click download or read online button to formal knot theory book pdf for free now. This report gives an overview of knot theory and some of its applications. A beginning for knot theory 1 exercises chapter 2. Figure 2 the trefoil as closed loop a knot presented in closed loop form is a robust object, capable of being pushed and twisted into. Download pdf formal knot theory free online new books in.
Since we have tried as much as possible to avoid formal terminol ogy, i. The author has provided a new supplement, entitled remarks on formal knot theory, as well as his article, new invariants in the theory of knots, first published in the american mathematical monthly, march 1988. In mathematics, the braid group on n strands denoted, also known as the artin braid group, is the group whose elements are equivalence classes of nbraids e. Knot theory is an exciting area of study, with many applications in the sciences.
A framed knot is the extension of a tame knot to an embedding of the solid torus d 2. Kauffman, formal knot theory, dover publications, 2006 this book. Although the subject matter of knot theory is familiar to everyone and its problems are easily stated, arising not only in many branches of mathematics but also in such diverse. Formal knot theory download formal knot theory ebook pdf or read online books in pdf, epub, and mobi format. An introduction to knot theory matt skerritt c9903032 june 27, 2003 1 introduction a knot, mathematically speaking, is a closed curve sitting in three dimensional space that does not intersect itself.
The merit of knot theory for the study of dimensional manifolds consists, first of all, in that every closed oriented dimensional manifold can be represented as a covering of the sphere, ramified over a certain link alexanders theorem. Virtual knots pdf download, detecting virtual knots pdf download and a selflinking invariant of virtual knots pdf download. Using the notion of a seifert surface of a knot, we define a knots genus, an additive invariant which allows to prove the existence of prime knots. For example, suppose the null hypothesis is that the wages of men and women are equal. Given a knot diagram, we color all the edges connecting the crossings by using three colors e.
Then, after defining an equivalence relation on all possible ways of factoring a knot, we will show that there is only one. Although the subject matter of knot theory is familiar. Foxcalculus lk notes on fox calculus, seifert pairing and alexander polynomial. The connected sum of two knots, k1 and k2, is formed. However, formatting rules can vary widely between applications and fields of interest or study. He is the editorinchief of jktr, editor of the series on knots and everything, full professor at uic and author of numerous books related to the theory of knots including knots and physics, knots and applications, on knots, and formal knot theory. In formally, an orientation of a knot can be thought of as a direction of travel around the knot. This paper is an introduction to the state sum model for the alexanderconway polynomial that was introduced in the the authors book formal knot theory.
In this chapter, we briefly explain some elementary foundations of knot theory. The author draws upon his work as a topologist to illustrate the relationships between knot theory and statistical mechanics, quantum theory, and algebra, as well as the role of knot theory in combinatorics. A 1tailed test typically requires a little more theory. Consider a knot j, and a second knot k created by going around the knot j n times, following the shape of j then k is the satellite knot with companion j the winding number of k is n theorem li, guo. An elementary introduction to the mathematical theory of knots. Intuitively if we were to take a piece of string, cord, or the like, tie a knot in it and then glue the loose ends together, we would have a knot. An introduction to knot theory knot theory knots, links. By the early 1990s, knot theory was recognized as another such area of mathematics, in large part due to c. In 1992, the journal of knot theory and its ramifications was founded. A quick trip through knot theory pdf file of the classic paper on knot theory by ralph fox 1961.
The main objective in knot theory is to invent more and more powerful invariants. In mathematical language, a knot is an embedding of a circle in 3dimensional euclidean space, r 3 in. Knot theory has many relations to topology, physics, and more recently. This paper has been a key introduction to knot theory for generations of knot theorists.
Knot theory, done with diagrams, illustrates the formalist idea very well. A knot is an embedding of the circle s 1 into threedimensional euclidean space r 3. There are no formal prerequisites, but some familiarity with linear and abstract algebra, as well as an. Duality conjecture in formal knot theory 233 the sum extending over all x.
Knot theory available for download and read online in other formats. Geometry of alternating links 32 exercises 40 chapters. Key concepts are related in easytoremember terms, and numerous helpful diagrams appear throughout the text. Introduction this paper is an introduction to the subject of virtual knot theory, a generalization of classical knot theory that i discovered in 1996 2. Thus a hyperbolic structure on a knot complement is a complete invariant of the knot. Since the early 1980s, graph theory has been a favorite topic for undergraduate research due to its accessibility and breadth of applications. Prathamesh prathamesh at imsc dot res dot in submission date. Formal knot theory is primarily about a reformulation of the alexander polynomial as a state summation. Although highdimensional knot theory does not have such glamorous applications as classical knot theory, it has many fascinating results of its own, which make use of a wide variety of sophisticated algebraic and geometric methods. An immediate invariant that comes to mind is the topological space s3 nk, the complement of knots.
The second is a list of engineering and scientific applications. An introduction to knot theory this essential introduction to vital areas of mathematics with connections to physics, while intended for graduate students, should fall within the ken of motivated upperdivision undergraduates. This is the rst book devoted entirely to highdimensional knot 1 see epple 68,69,70 for the history of knot. Kauffman this paper is an introduction to the theory of virtual knots. Download pdf formal knot theory free online new books. After discussing the history of the subject and covering basic definitions, well discuss the property of tricolorability and prove its use in answering a fundamental question in knot theory. Knot theory was a respectable if not very dynamic branch of topology until the discovery of the jones polynomial 1984 and its connections with physics speci cally, quantum eld theory, via the work of witten. Motivation for such a topological study of knots is meant to come from a curiosity to know how the ge ometry of threedimensional space can be explored by knotting phenomena using precise mathematics. An introduction to knot theory and the knot group 5 complement itself could be considered a knot invariant, albeit a very useless one on its own. The concepts that were formalized include definitions of tangles, links, framed links and linktangle equivalence. The trefoil knot shown in figure 2 is an example of such a closed knotted loop. This work was essentially a compilation of the knowledge at the time. For more papers on virtual knot theory by kauffman and others, please browse the arxiv.
The author has provided a new supplement, entitled remarks on formal knot theory, as well as his article, new invariants in the theory of knots, first published in the american. Knot theory this chapter looks at some of the fundamental building blocks of knot theory. We have seen in this chapter the formal definitions of hopf algebras. Although these do have a signi cant in uence on elementary knot theory, via unknotting number. Click download or read online button to formal knot theory book pdf. This work contains a formalization of some topics in knot theory. An international conference on knot theory and related topics was held at osaka as a satellite conference of icm kyoto in 1990. A knot is called prime if it can not be represented as a connected sum of two knots such that both of these are knotted. The first is that knot theory is a treasure chest of examples for several different branches of topology, geometric group theory, and certain flavours of algebra. The knot group of a knot awith base point b2s3 ima is the fundamental group of the knot complement of a, with bas the base point. Knot diagrams and their in variants, braid groups, seifert forms, cyclic covers of knot complements and their invariants, jones type polynomial invariants, quantum invariants, surfaces in knot complements, hyper.
In knot theory and 3manifold theory, often the adjective tame is omitted. The duality conjecture in formal knot theory core reader. Knot theory was a respectable if not very dynamic branch of topology until the discovery of the jones polynomial 1984 and its connections with physics speci. The knot theoristos usual convention for preventing this is to assume that the knot is formed in a closed loop of string. Knot theory and its applications ams tesi di laurea.
The course is assessable to advanced undergraduate students. Kauffman draws upon his work as a topologist to illustrate the relationships between knot theory and statistical. The ideas used in this chapter can be found in most introductory books or courses on. Knot logic and topological quantum computing with majorana fermions louis h. Knot theory weeks 110 ucsb 2015 outside of mathematics, knots are ways to loop a single piece of string around itself. For more papers on virtual knot theory by kauffman and others, please browse the arxiv on the web. There are no formal prerequisites, but some familiarity with linear and abstract algebra, as well as an ability to visualize objects in three dimensions is useful. Knot theory now plays a large role in modern mathematics, and the most signifi. The framing of the knot is the linking number of the image of the ribbon i. If we could completely understand hyperbolic structures on knot complements, we could completely classify hyperbolic knots. Jan 01, 2006 key concepts are related in easytoremember terms, and numerous helpful diagrams appear throughout the text.