Introduction to knot theory eth spring semester 2015 books that can be used in secondary school gymnasium. The commonest way of presenting a specific knot to the human eye is by a diagram of the type shown in fig. Knot theory is an appealing subject because the objects studied are familiar in everyday physical space. It is valuable as a professional reference and will serve equally well as a text for a course on knot theory. Knot theory the website for learning more about knots.
After discussing the history of the subject and covering basic definitions, well discuss the property of tricolorability and prove its use in answering a. Introduction to knot theory, fs 2019 meike akveld bibliography. 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. It describes knot coloring 2 by quandles, the fundamental group of a knot complement, and handledecompositions of 3 knot complements. 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. Pdf history of knot theory jozef przytycki academia. Deltagroupoids in knot theory unsolved problems in virtual knot theory and combinatorial knot. We have also avoided 4dimensional questions, such as the sliceribbon conjecture problem 1. 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. A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closedthere are no ends to tie or untie on a mathematical knot. A rough outline that we may or may not adhere to is the following.
An introduction to knot theory graduate texts in mathematics. Knot theory is the study of smooth simple closed curves in 3dimensional space. Knot theory by charles livingston cambridge university press. Peter guthrie tait was the first person to make charts describing mathematical knots in the 1860s. Knots which are not tame are called wild, and can have pathological behavior. Etnyre knot spinning greg friedman the enumeration and classification of knots and links jim hoste knot diagrammatics louis h. These notes summarize the lectures delivered in the v mexican school of particle physics, at the university of guanajuato. Applications of planar graphs in knot theory by david. Thistlethwaite, editors 1 colin adams, hyperbolic knots 2 joan s. The simplest common knots which can be tied using a string are the overhand knot and the figure eight knot. A tame knot is any knot equivalent to a polygonal knot. In knot theory, the ends of the rope are attached so that there is no possible way for the knot to be untied. It will show how knot theory originated from early attempts to apply the study of knots to other disciplines, and over time has emerged as a promising field of mathematical exploration in its own right. The knot atlas wiki, by dror barnatan and scott morrison.
In these terms, the knot theory of complex plane curves is solidly part of modern knot theory the knots and links in question are clinks, and the extra structures variously algebraic. Classical knot theory, in particular, is concerned with the ways in which a circle or a disjoint union of circles can be embedded in r3. Fundamental concepts of knot theory 5 1 the elementary knot moves 6 2 the equivalence of knots i 7 3 tie equivalence of knots ii 9 4 ltnfcs 14 5 knot decomposition and the semigroup of a knot 17 6 the cobordism group of knots 23 knot tables 25 1 regular diagrams and alternating knots 26 2 knot tables 30 3 knot graphs 34. In knot theory and 3manifold theory, often the adjective tame is omitted. I guess there should be some, since proteins are quite long and probably some of their properties are related whether they are knotted or not.
This is a very accessible topic and there are plenty of readable books. This paper will provide the reader with an accessible route that begins with basic knot theory and leads into in. Two applications of elementary knot theory to lie algebras. Applications of planar graphs in knot theory reidemeister moves on graphs knot to planar graph conversion kcircuit kcircuit seifert algorithm almost planar seifert surfaces seifert surface heres one way kcircuit on k3 appropriated from sharon goldwaters web page. Knots can be studied at many levels and from many points. In the present book, we give both the old theory of knots, such as the fun damental group, alexanders polynomials, the results of dehn, seifert, burau, and. A brief history of knot theory erin col berg abstract.
Resources for undergraduate knot theory mathematics. Lecture notes class projects course information this is an introductory course in knot theory. 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. Applications of the knot theory to biologypharmacology. Knot theory consists of the study of equivalence classes of knots. The chapter then describes the basics of higherdimensional knot theory and then turns to specific constructions, including simple spinning, superspinning, twist spinning, frame spinning, rolling, and deform spinning. Knot theory this chapter looks at some of the fundamental building blocks of knot theory. The ideas used in this chapter can be found in most introductory books or courses on. Knots, quandles, fundamental groups, handles, knot colorings, symmetry, 5 surfaces, klein bottle, projective plane 6 1. Course information this is an introductory course in knot theory. The function j12 x plays an important role in lie theory. The book is an excellent exposition on knot theory. Given a tangled loop of string, is it really knotted or can it, with.
The author glosses over many technical details, but that allows the reader to delve more deeply into the material. Charles livingston knot theory chapter 3 sections 2 through 4. Jan 04, 2011 this paper is an introduction to virtual knot theory and an exposition of new ideas and constructions, including the parity bracket polynomial, the arrow polynomial, the parity arrow polynomial. What can be included in the course will depend a lot on the background of the students. For more papers on virtual knot theory by kauffman and others, please browse the arxiv on the web. Knot theory, a lively exposition of the mathematics of knotting, will appeal to a diverse audience of mathematical readers, from undergraduates to professionals. Tricolorability of knots massachusetts institute of. Number of crossings of a knot the crossing number of a knot class is the minimum crossing number over all of the knots in the class open conjecture.
Charles livingston, knotentheorie fur einsteiger, viewegverlag, 1995. Knot theory is notable not only for its expert presentation of knot theorys state of the art but also for its accessibility. So students were able to see the point of the course. The concordance genus of knots 1 introduction and basic. While exploring the mathematical properties of tangles you will find connections to popular commuter games and an unsolved problem worth a million dollars. A knot is an embedding of the circle in the euclidean plane r2. In mathematics, a knot is an embedding of a circle s 1 in 3dimensional euclidean space, r 3 also known as e 3, considered up to continuous deformations. Also it has chapters on the recent applications of knot theory to other fields such as physics, chemistry and biology. We include examples of alexander polynomial one knots for which the invariant is nontrivial, including all iterated untwisted positive doubles of knots with nonnegative thurstonbennequin number, such as the trefoil, and explicit.
This book is a survey of current topics in the mathematical theory of knots. Virtual knots pdf download, detecting virtual knots pdf download and a selflinking invariant of virtual knots pdf download. Table of knot invariants, by charles livingston and jae choon cha. Louis kauffman discusses applications of knot theory to physics, nadrian seeman discusses how topology is used in dna nanotechnology, and jonathan simon discusses the statistical and energetic properties of knots and their relation to molecular biology. Department of mathematics schnupperkurs on knot theory kit. Discovering the art of knot theory lets you, the explorer, investigate the mathematical concepts and ideas of knot theory using tangles. Journal of knot theory and its ramifications, request pdf.
Livingston, knot theory, the mathematical association of amer ica inc. One thing that makes knot theory so interesting for mathematicians today is the fact that its such a new topic knot theory is a relatively young field with many opportunities for discovery and exploration by mathematicians young and old. If we could completely understand hyperbolic structures on knot complements, we could completely classify hyperbolic knots. Knot theory, 1993 accessible, but too pedestrian for my taste p. A list of recommended books in topology cornell university. Abstract we consider oriented knots and links in a handlebody of genus g through appropriate braid representatives in s3, which are elements of the.
A survey 3 john etnyre, legendrian and transversal knots 4 greg friedman, knot spinning 5 jim hoste the enumeration and classi. It then discusses some new hybrid constructions, such as frame twist spinning, frame deform spinning. The combinatorial revolution in knot theory sam nelson k not theory is usually understood to be the study of embeddings of topological spaces in other topological spaces. Kauffman a survey of classical knot concordance charles livingston knot theory of complex plane curves lee rudolph thin position. Livingston knot theory, maa carus monograph 24 shows a righthanded 3,5 and a lefthanded 3,5 torus knot on page 4.
An elementary introduction to the mathematical theory of knots. Knot theory manuela almeida applied mathematics and computation, ist february 8, 2012 1 introduction in mathematics, knot theory is the study of knots. Most of the material is from livingston 3 and adams 1. 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. Cambridge core geometry and topology knot theory by charles livingston. Dna is the genetic material of all cells, containing coded information about cellular molecules and processes.
The second half of this volume is focused on three particular applications of knot theory. Rebecca hardenbrook a knot vices guide to untangling knot theory 3 26. What are the applications of the knot theory to biologypharmacology. Using the notion of a seifert surface of a knot, we define a knot s genus, an additive invariant which allows to prove the existence of prime knots. 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. Peter cromwell cambridge university press, 2004 another good introduction to knot theory, more mathematical than the one by livingston often more technical than the style of this lecture will be. 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.
Then we will go a little deeper into the theory of categori. When graph theory meets knot theory denison university. An elementary introduction to the mathematical theory of knots, w. This course will cover at least chapters 16 and 10 of livingston, plus additional material as time permits. Some of the material will be used later in this report, while some is included simply to give an idea of di erent techniques used in knot theory. The purpose of this course is to learn the basics of knot theory. In mathematical language, a knot is an embedding of a circle in 3dimensional euclidean space, r 3 in. Math 148 knot theory course information fall 2012 time and.
Dna consists of two polynucleotide strands twisted around each other in a double helix. The study of knots and links begins with simple intuitive problems but quickly leads to sophisticated mathematics. 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. Prerequisites include multivariable calculus, linear algebra, and a proof course, but no exposure to topology or abstract algebra. In topology, knot theory is the study of mathematical knots. It is an essential tool in the study of 3 and 4dimensional manifolds. We give a survey of the application of ashtekars variables to the quantization of general relativity in four dimensions with special emphasis on the application of techniques of analytic knot theory to the loop representation. The third chapter, called invariants of knots, does not deal with knot polynomials as the expert would expect, but with three extremely simple but hopelessly uncomputable from the practical4.
In this chapter, we briefly explain some elementary foundations of knot theory. Knot theory is a rapidly developing field of research with many applications, not only for mathematics. Daniel kleitman massachusetts institute of technology abstract. Knot theory is an exciting area of study, with many applications in the sciences. Although these do have a signi cant in uence on elementary knot theory, via unknotting number. Its square, jx, is the jacobian of the exponential mapping from g to the lie group g.
By the early 1990s, knot theory was recognized as another such area of mathe. Knot theory is a large and active research area of mathematics that employs advanced techniques of abstract algebra and geometry. Knots and physics world scientific, 1991 a large but very readable book on theory. Thus a hyperbolic structure on a knot complement is a complete invariant of the knot. What is perhaps most surprising is that many problems that are intractable in the classical case have been solved for highdimensional knots. The present volume, written by a wellknown specialist, gives a complete survey of this theory from its very beginnings to todays most recent research results. This paper will provide a chronological description of the development of knot theory. Also we will give a short introduction to knot theory and give a description of the most important knot invariants. Then, after defining an equivalence relation on all possible ways of. I am going to teach a 400level topics class on knot theory at an american, midsized, public university. The enumeration and classification of knots and links 209 j. This account is an introduction to mathematical knot theory, the theory of knots and links of simple closed curves in threedimensional space.
Knot theory of complex plane curves 349 l rudolph 9. Among other info, it contains rolfsens table of knots up to 10 crossings. Algebraic concordance and higher dimensional knot theory. This paper is a very brief introduction to knot theory. It describes knot coloring by quandles, the fundamental group of a knot complement, and handledecompositions of knot complements. For a mathematician, a knot is a closed loop in 3dimensional space. The following ones are listed more or less in order of increasing sophistication. Charles livingston, indiana university, bloomington.
The author introduces tools from linear algebra and basic group theory and uses these to study the properties of knots, highdimensional knot theory and the conway, jones and kauffman polynomials. This report gives an overview of knot theory and some of its applications. 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. 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. Introduction to knot theory, spring semester 2015 meike akveld bibliography. Introduction to knot theory eth fs 2019 books that can be used in secondary school \gymnasium. 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. An indispensable book for everyone concerned with knot theory. A polygonal knot is a knot whose image in r3 is the union of a finite set of line segments. The advantage i found in teaching knot theory, as against say homology theory, was that the basic problems could be stated at the beginning, and some methods were given, relating as said above to other nice mathematics, for some measure of solution. Since the early 1980s, graph theory has been a favorite topic for undergraduate research due to its accessibility and breadth of applications.
Livingston talks not about the number of 3colourings but about 3colourability. Although the subject matter of knot theory is familiar. Knot theory was a respectable if not very dynamic branch of topology. Graduate students beginning a program of advanced study will find a worthwhile overview. Time for all that glitters, sparkles, blinks and sings or dances at the flip of a switch, to be. It assumes only a general background in mathematics yet contains a great deal to occupy even the expert. Besides, the braid word given in the properties section applies to a p,q not p,q torus knot, because the braid generators are left twists. Table of contents for the handbook of knot theory william w. Braids, links and mapping class groups, by joan birman. Knot theory in handlebodies reinhard haringoldenburg and sofia lambropoulou. The concepts and practice of knot theory are very well presented.
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