Although these do have a signi cant in uence on elementary knot theory, via unknotting number. Knots can be studied at many levels and from many points. The following ones are listed more or less in order of increasing sophistication. Claude monet a knot is a circle smoothly embedded in 3dimensional euclidean space or its compacti cation. Charles livingston knot theory chapter 3 sections 2 through 4. The study of knots and links begins with simple intuitive problems but quickly leads to sophisticated mathematics.
A knot is a tangled string in euclidean 3space r3 which is usually considered as a closed tangled string in r3, and a link is the union of some mutually disjoint knots see figure 1. This paper will provide the reader with an accessible route that begins with basic knot theory and leads into in. Ellie obrien knot theory and its applications april 26, 2016 3 4. Knot theory, 1993 accessible, but too pedestrian for my taste p. This account is an introduction to mathematical knot theory, the theory of knots and links of simple closed curves in threedimensional space. The rings are unique and very comfortable to wear, even for working hands. Knot theory, 1993 accessible, but too pedestrian for.
We have enjoyed their silicone rings for several years. A knot is a single circle embedded in the affine space of dimension 3 as a smooth curve not crossing itself. A tame knot is any knot equivalent to a polygonal knot. Knot theory by charles livingston cambridge university press. I also have to mention knot theory s excellent customer service and communications with the customer. In mathematics, we mean something slightly di erent by a knot.
The concepts and practice of knot theory are very well presented. 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. Introduction to knot theory, spring semester 2015 meike akveld bibliography. Knot theory, second edition is notable not only for its expert presentation of knot theorys state of the art but also for its accessibility. This report gives an overview of knot theory and some of its applications. Pictures of knots with 10 crossings taken from the tables of tate and little. 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. It describes knot coloring by quandles, the fundamental group of a knot complement, and handledecompositions of knot complements.
Knot theory, a lively exposition of the mathematics of knotting. The knotplot site some knot theory books that you may find in the library. In this chapter, we briefly explain some elementary foundations of knot theory. 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. Abstract we consider oriented knots and links in a handlebody of genus g through appropriate braid representatives in s3, which are elements of the. In knot theory and 3manifold theory, often the adjective tame is omitted. This paper will provide the reader with an accessible route that begins with basic knot theory and leads into in teresting realms of moder research.
The knot atlas wiki, by dror barnatan and scott morrison. They are somewhat rough, but possibly of interest to others. Knot theory is notable not only for its expert presentation of knot theorys state of the art but also for its accessibility. Close to what well cover in the first half of the course. Kauffman this paper is an introduction to the theory of virtual knots. The author glosses over many technical details, but that allows the reader to delve more deeply into the material. An aid to understanding rational2bridge knot construction and notation. 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. Knot theory in handlebodies reinhard haringoldenburg and sofia lambropoulou. Thesenotes consist oflectures froma graduate courseitaught at brigham young university in winter semester 2010. A list of recommended books in topology cornell university. This is a very accessible topic and there are plenty of readable books. This paper will provide a chronological description of the development of knot theory. Rebecca hardenbrook a knotvices guide to untangling knot theory 25 26.
Introduction to knot theory eth spring semester 2015 books that can be used in secondary school gymnasium. Livingston, knot theory, the mathematical association of amer ica inc. A brief history of knot theory erin col berg abstract. In the mathematical field of knot theory, a knot invariant is a quantity in a broad sense defined for each knot which is the same for equivalent knots. Graduate students beginning a program of advanced study will find a worthwhile overview, and the reader will need no training beyond linear algebra. Knot theory weeks 110 ucsb 2015 outside of mathematics, knots are ways to loop a single piece of string around itself. 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. Many knot invariants are known and can be used to distinguish knots. I list below several books which are perhaps the closest to the topics we will study in class and are available at the ucla library. Table of knot invariants, by charles livingston and jae choon cha.
The concordance genus of knots 1 introduction and basic. Virtual knots pdf download, detecting virtual knots pdf download and a selflinking invariant of virtual knots pdf download. An elementary introduction to the mathematical theory of knots colin c. 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. This paper will provide the reader with an accessible route that begins with basic knot theory and leads into interesting realms of modern research. This paper is a very brief introduction to knot theory. Start moving along the knot, as we encounter the crossings we start numbering them, so every crossing has two numbers assigned to it once we have traced the entire knot. 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. For more papers on virtual knot theory by kauffman and others, please browse the arxiv on the web. The euler characteristic for a 1dimensional object is 0 when applied to a knot, hence that does not lead to a distinction. Knot theory high performance silicone wedding rings for man. 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 theory is the embedding of one topological space into another topological space. A knot is an embedding of the circle in the euclidean plane r2.
Algebraic concordance and higher dimensional knot theory. Knot equivalence is in general a difficult thing to prove, as well see shortly. Some fundamental theorems of involved in knot theory include one involving reidemeister moves transforming knots into unknots and knot. Knot theory mathematical association of america textbooks 9780883850275. Show less this book is a survey of current topics in the mathematical theory of knots. Knot theory consists of the study of equivalence classes of knots. An elementary introduction to the mathematical theory of knots, w.
The genus of a knot, introduced by seifert 12, is another classic invariant in knot theory. I am going to teach a 400level topics class on knot theory at an american, midsized, public university. Thistlethwaite, editors 1 colin adams, hyperbolic knots 2 joan s. Cambridge core geometry and topology knot theory by charles livingston. Also, as we prove, not all conjugations in the groups bg.
We have also avoided 4dimensional questions, such as the sliceribbon conjecture problem 1. Knot theory is of central importance in pure and applied mathematics, as it stands at a crossroads of topology, combinatorics, algebra, mathematical physics and biochemistry. A reidemeister move is an operation that can be performed on the diagram of a knot whithout altering the corresponding knot. Knot theory is now believed that a scientific study to be associated with the atomic theory of vortex atoms in ether around the end of the nineteenth century.
For more papers on virtual knot theory by kauffman and others, please browse the arxiv. We often receive compliments and questions about them. Rebecca hardenbrook a knotvices guide to untangling knot theory 3 26. Knot theory was rst proposed as the awed vortex model of atoms by lord kelvin, but was later proved to be incorrect. Charles livingston, indiana university, bloomington. Knot theory is dealing with embedding of curves in manifolds of dimension 3.
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. A polygonal knot is a knot whose image in r 3 is the union of a finite set of line segments. Knot theory in handlebodies 3 a twist has to take place anywhere on the right of the braid. Knot theory ams bookstore american mathematical society.
It is valuable as a professional reference and will serve equally well as a text for a course on knot theory. Two diagrams represent equivalent knots if and only if one can be transformed into the other by a nite sequence of reidemeister moves. Combinatorial methods in knot theory professor josh greene boston college january 16, 20 1 foundations what is a knot. An introduction to knot theory, by raymond lickorish. An elementary introduction to the mathematical theory of knots.
Table of contents for the handbook of knot theory william w. Ozsvath and szabo have defined a knot concordance invariant. Knots which are not tame are called wild, and can have pathological behavior. The book closes with a discussion of highdimensional knot theory and a presentation of some of the recent advances in the subject the conway, jones, and kauffman polynomials. Knot groups will be developed here using combinatorial methods. 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. Knot theory is an exciting area of study, with many applications in the sciences. 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. Knotted dna and a dna page from the knots exhibition at bangor. Resources for undergraduate knot theory mathematics. Most of the material is from livingston 3 and adams 1.
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. Prerequisites include multivariable calculus, linear algebra, and a proof course, but no ex. Williams may, 2016 1 introduction color is my daylong obsession, joy and torment. Graduate students beginning a program of advanced study will find a worthwhile overview, and the. But it will be critical in working towards the solution of one of the central problems of knot theory. A survey 3 john etnyre, legendrian and transversal knots 4 greg friedman, knot spinning 5 jim hoste the enumeration and classi. Knot theory and its applications occidental college. The equivalence is often given by ambient isotopy but can be given by homeomorphism. A supplementary section presents the fundamental group, which is a centerpiece of algebraic topology. The field of mathematics called algebraic topology is devoted to developing and exploring connections between topology and algebra. I also have to mention knot theorys excellent customer service and communications with the customer. Knot theory high performance silicone wedding rings for. Seifert therefore used a connected, oriented, compact surface that has the knot as its boundary to dene the genus of a knot. Among other info, it contains rolfsens table of knots up to 10 crossings.