This ar ticle is an intr oduction to relationships betwe en knot the or y and the oretic al physics. 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. Introduction knot theory is a theory studying the macroscopic physical phenomena of strings embedded in threedimensional space. This invaluable book is an introduction to knot and link invariants as generalized amplitudes for a quasiphysical process. W e give an exp osition of the the or y of polynomial invar iants of knots and links, the w itten functional inte gral for mulation of knot and link invar iants, and the beginnings of top olo gic al quantum. Knot theory is used to solve fundamental questions in physics ranging from. It is a consequence of hakens work that the genus of a knot is algorithmically computable, because a minimal genus seifert surface can be arranged to be a fundamental normal surface. Prerequisites include multivariable calculus, linear algebra, and a proof course, but no exposure to topology or abstract algebra. In mathematical language, a knot is an embedding of a circle in 3dimensional euclidean space, r 3 in. Jun 30, 2017 hello, does anybody know of a book that has to do with topology and particularly with knot theory and their applications to condensed matter physics.
As michael comments, colin adams has a well regarded text called the knot book. This book is an introduction to hyperbolic geometry in dimension three, and its applications to knot theory and to geometric problems arising in knot theory. How ever, thurston and gabai found another method for determining the genus of knots, by using the theory of taut foliations 97, 21. Discovering the art of knot theory lets you, the explorer, investigate the mathematical concepts and ideas of knot theory using tangles. The first part covers basic tools in hyperbolic geometry and geometric structures on 3manifolds. Knot complements and groups, topology 26 1987 4144. Knot theory and its applications kunio murasugi springer. Physical knot theory is the study of mathematical models of knotting phenomena, often motivated by considerations from biology, chemistry, and physics kauffman 1991.
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. The demands of knot theory, coupled with a quantum statistical framework, create a context that naturally and powerfully includes an extraordinary range of interrelated topics in topology and mathematical physics. The idea of topologically stable matter has rst been proposed by lord kelvin. From around 1980, knot theory came to attention not only in almost all areas of mathematics, but also in the fields of science that will be cuttingedge researches, such as gene synthesis, quantum statistical mechanics, soft matter physics, biochemistry, polymer network, applied chemistry. Spontaneous knotting of an agitated string, pnas 2007. I am going to teach a 400level topics class on knot theory at an american, midsized, public university. This book, which is an elaboration of a series of lectures given by fox at haverford college while a philips visitor there in the spring of 1956, is an attempt to make the subject accessible to. A fundamental problem in knot theory is determining when two knots are the same, which leads to the study of knot invariants, such as knot polynomial. 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.
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. 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. The most striking manifestation of this fact is the vast richness of novel. A supplementary section presents the fundamental group, which is a centerpiece of algebraic topology. 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. He also presents the remarkable applications of knot theory to modern chemistry, biology, and physics. 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 theory now plays a large role in modern mathematics, and the most signifi cant results in this. Here, however, knot theory is considered as part of geometric topology. Braid group, knot theory and statistical mechanics ii. Jan 01, 1996 this book is directed to a broad audience of researchers, beginning graduate students, and senior undergraduate students in these fields.
This comic book comes with a plastic rope that can be knotted, unknotted, and twisted into different shapes. The book contains most 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. Request pdf handbook of knot theory this book is a survey of current topics. As a subfield of topology, knot theory forms the core of a wide range of problems dealing with the position of one manifold imbedded within another. Over the last fifteen years, the face of knot theory has changed due to various new theories and invariants coming from physics, topology, combinatorics and algebra. Its very humorous but is a genuine introduction to the mathematics involved. Thousands of refereed articles about knots have been published during just the past ten years. Introduction the topological objects have been assuming increasingly important roles in physics. Scientists wit h primar y interest s i n physic s o r biochemistr y shoul d find the. Mathematicians link knot theory to physics the new york. In quantum physics, a knot may be regarded as the orbit in spacetime of a charged particle. Physical knot theory is used to study how geometric and topological characteristics of filamentary structures, such as magnetic flux tubes, vortex filaments, polymers, dnas, influence their physical properties and. Scottish physics and knot theorys odd origins introduction. In fact, one of the attractions of knot theory is its extensive interactions with many di erent branches of mathematics.
Colin adams engages the reader with fascinating examples, superb figures, and thoughtprovoking ideas. In the rst chapter, we will see how the research on this subject changed during the time. This book is directed to a broad audience of researchers, beginning graduate students, and senior undergraduate students in these fields. Knot theory is a kind of geometry, and one whose appeal is very direct hecause the objects studied areperceivable and tangible in everydayphysical space.
It suffices to mention the great progress in knot homology theory khovanov homology and ozsvathszabo heegaardfloer homology, the apolynomial which give rise to strong invariants of knots and 3manifolds, in particular. 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. This book is based on the 1989 volume but has new material included and new contributors. Knot theory is one of the most active research areas of mathematics today. The knot book is also about the excitement of doing mathematics. Resources for undergraduate knot theory mathematics. Knot theory is a concept in algebraic topology that has found applications to a variety of mathematical problems as well as to problems in.
Knot theory is a rapidly developing field of research with many applications, not only for mathematics. Adams has also written a comic book about knot theory called why knot. The present volume is an updated version of the book edited by c n yang and m l ge on the topics of braid groups and knot theory, which are related to statistical mechanics. 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 book is directed to a broad audience of researchers, beginning graduate. Sections 6 and 7 provide an introduction to vassiliev invariants and the remarkable relationship between lie algebras and knot theory. The author takes a primarily combinatorial stance toward. 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.
Below, edward witten, charles simonyi professor in the school of natural sciences, describes the history and development of the jones polynomial and his. In the twentieth century, mathematicians developed a deep theory of knots, which was revolutionized by the discovery of the jones polynomiala way to calculate a number for every knotby vaughan f. Research that makes people laugh and then think 2008 physics prize. One publication, journal of knot theory and its rami. Alexander introduced algebraic methods into the theory, and the rst book about knots, knotentheorie was published by k. The demands of knot theory, coupled with a quantumstatistical framework, create a context that naturally and powerfully includes an extraordinary range of interrelated topics in topology and mathematical physics. This book is an introduction to knot and link invariants as generalized amplitudes vacuumvacuum amplitudes for a quasiphysical process. The researchers say that the connection was unexpected because particle physics seemed far removed from knot theory, a branch of topology, the study of the properties of space and shapes. They can be admired as artifacts of the decorative arts and crafts, or viewed as accessible intimations of a geometrical sophistication that may never be attained. In this dissertation, we are going to give a brief introduction of knot theory, looking at di erent aspects. By 1970, knot theory had become a welldeveloped area of topology 7. Knots may be regarded as formed by interlacing and looping a piece of string in any fashion and then joining the ends. Pdf knots are usually categorized in terms of topological properties that are invariant.
Book on knot theory in condensed matter physics physics forums. But to use the chernsimons function, the knot must be a path in a spacetime of three dimensions two space dimensions and one time. The discovery of the jones polynomial by vaughan jones in 1984 not only showed a connection between knot theory and di erent areas of mathemat ics operator algebras, braid theory, quantum groups, but also to physics statistical models 2, 8. One way of calculating the jones polynomial in quantum theory involves using the chernsimons function for gauge fields. In this way knots have become a testing ground not only for topological ideas but also for the methods of modern theoretical physics. 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.
In the present book, we give both the old theory of knots, such as the fun damental group. The knot book is an introduction to this rich theory, starting with our familiar understanding of knots and a bit of college algebra and finishing with exciting topics of current research. Throughout ths book, we shall concentrate on lucid exposition, and. I was looking at baezs gauge fields, knots and gravity and i was wondering if theres anything like it for condensed matter physics. 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. Physical knot theory is used to study how geometric and topological characteristics of filamentary structures, such as magnetic flux tubes, vortex filaments, polymers, dnas, influence their physical properties and functions. The first question that arises is whether such a curve is truly knotted or can simply be untangled. Knots can be studied at many levels and from many points of view. Knots and quantum theory institute for advanced study.
This account is an introduction to mathematical knot theory, the theory of knots and links of simple closed curves in threedimensional space. Knot theory, in mathematics, the study of closed curves in three dimensions, and their possible deformations without one part cutting through another. Hello, does anybody know of a book that has to do with topology and particularly with knot theory and their applications to condensed matter physics. Dorian raymer of the ocean observatories initiative at scripps institution of oceanography, usa, and douglas smith of the university of. In topology, knot theory is the study of mathematical knots. The demands of knot theory, coupled with a quantum statistical framework, create a context that naturally and powerfully includes an extraordinary range of interrelated topics in topology and mathematical. Knot theory consists of the study of equivalence classes of knots. While exploring the mathematical properties of tangles you will find connections to popular commuter games and an unsolved problem worth a million dollars. Book on knot theory in condensed matter physics physics. He describes the making of a knot with 7 bights and 8 leads.
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