These are called links, or in the case of just one component knots. We survey some tools and techniques for determining geometric properties of a link complement from a link diagram. If you want a book with references on many aspects of knot theory, then this is a very good book. This book is a survey of current topics in the mathematical theory of knots. Twisted quandles a survey of quandles with some recent developments mohamed elhamdadi university of south florida qqq 2016 mohamed elhamdadi a survey of quandles with some recent developments.
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. However, burde and zieschang have written an essential reference for those wanting to do research in knot theory or 3manifold theory. A survey of applications of surgery to knot and link theory. Table of contents for the handbook of knot theory william w. A crossing a knot is in the threedimensional space and we think that it is made of a very thin string. We always require twist regions to be alternating otherwise, d can be simpli. The main purpose is to give a short introduction to the subject and a guide to the applications that have been found thus far for quandle cocycle invariants. Quandle cocycle invariant of knots quandle modules and generalized quandle cohomology theory some generalizations of quandles. The minimum of td over all diagrams of k is denoted tk. Table of contents preface ix a prelude to the study of knot theory xi notes on research conventions and notations xxi chapter 0 fundamentals of knot theory 0. Given a knot in the 3sphere, the knot complement is all the points of the 3sphere not contained in the knot. Elementary constructions of homfly and kau man polynomials l. The topics include alexander polynomials, jones type polynomials, and vassiliev invariants. For a mathematician, a knot is a closed loop in 3dimensional space.
The first chapter introduces the path which led to the study of the mathematical theory of knots and a brief history of the subject with the relevance of research in it. Not an easy read for someone who has no prior knowledge of knot theory. A survey of hyperbolic knot theory 5 an equivalence class. A major theorem of gordon and luecke states that at most two knots have homeomorphic complements the original knot and its mirror reflection. A survey of knot theory, 1990 a lot of material, but quite concise v. The present volume, written by a wellknown specialist, gives a complete survey of this theory. Pdf survey on recent invariants on classical knot theory. Generally, knot theory deals with onedimensional closed piecewise linear submanifoldsof the threedimensional sphere s3, in other wordswe consider the images of piecewise linear embeddings of disjoint unions of circles into s3.
This article attempts a rapid survey of the role of surgery in the development of knot and link theory. A reidemeister move is an operation that can be performed on the diagram of a knot whithout altering the corresponding knot. It is a meeting ground of such diverse branches of mathematics as group theory, matrix theory, number theory, algebraic geometry, and differential geometry, to name some ofthe more prominent ones. A supplementary section presents the fundamental group, which is a centerpiece of algebraic topology. By the early 1990s, knot theory was recognized as another such area of mathe. Thenumberoftwistregionsinaprime,twistreduceddiagramisthetwistnumber of the diagram, and is denoted td. I am going to teach a 400level topics class on knot theory at an american, midsized, public university. The analysis of this earthquake curve can be considered as knot theory in the wide sense. 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. We have also avoided 4dimensional questions, such as the sliceribbon conjecture problem 1. Knot theory is a rapidly developing field of research with many applications not only.
A survey of knot theory, by akio kawauchi, birkhauserverlag, basel. This survey is primarily devoted to three main topics. Knot theory is a kind of geometry, and one whose appeal is very direct hecause the objects studied areperceivable and tangible in everydayphysical space. These notes summarize the lectures delivered in the v mexican school of particle physics, at the university of guanajuato. Knot theory, 2004 a lot of material, but quite concise reidemeister. Since the early work of tait, knot theory has been in. If you have just started studying knot theory or want to study, dont buy it unless you find it at a very low price like i did.
Knot theory is a rapidly developing field of research with many applications not only for mathematics. Our main goal is to focus on the methods, techniques, and tools of the field, in the hopes that this paper will lead to more research, rather than strictly listing previous results. In particular, we survey the tools used to estimate geometric invariants in terms of basic diagrammatic link invariants. Since the early 1980s, graph theory has been a favorite topic for undergraduate research due to its accessibility and breadth of applications.
Knot theory is a subject with a deep cultural background and a short mathe matical history. In topology, knot theory is the study of mathematical knots. 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. In dimension one, the manifold embedding problem is knot and link theory.
The present volume, written by a wellknown specialist, gives a complete survey of knot theory from its very beginnings to todays most recent research results. Survey on recent invariants on classical knot theory. Although these do have a signi cant in uence on elementary knot theory, via unknotting number. In addition to knot theory, quandles have found applications in other areas which are only mentioned in passing here. This in effect turns the study of knots into the study of their complements, and in turn into 3manifold theory.
A brief survey of the vast field of knot theory is given in this final report named knot 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. Virtual knots pdf download, detecting virtual knots pdf download and a selflinking invariant of virtual knots pdf download. Covers also some general 3manifold theory relevant to knot theory. Corrections on the table of data appendix f of a survey. This is a survey of the impact of thurstons work on knot theory, laying emphasis on the two characteristic features, rigidity and flexibility, of 3dimensional hyperbolic structures. As an example of the sort background required, i sample the following instances of nontrivial topology.
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. With one exception, all of the results presented in this survey have appeared elsewhere in the literature. Kawauchi, editor, a survey of knot theory, birkhauser. For all of these results, we point to references rather than giving rigorous proofs. Thistlethwaite, editors 1 colin adams, hyperbolic knots 2 joan s. Resources for undergraduate knot theory mathematics. A survey of knot theory birkhauser verlag basel boston berlin.
However, we often include quick sketches of arguments to convey a sense of. 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. A survey of quandle ideas introductory lectures on knot. Not only has knot theory developed and grown over the years in its own right, but also the actual mathematics of knot theory has been shown to have. For more papers on virtual knot theory by kauffman and others, please browse the arxiv.
Knot theory is a rapidly developing field of research with many applications not. When graph theory meets knot theory denison university. Pdf a survey of hyperbolic knot theory semantic scholar. Request pdf a survey of hyperbolic knot theory we survey some tools and techniques for determining geometric properties of a link complement from a link diagram.
A survey of the impact of thurstons work on knot theory. An indispensable book for everyone concerned with knot theory. Knot theory is a rapidly developing field of research with many applications, not only for mathematics. 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.