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High-dimensional knot theory is the study of the embeddings of n-dimensional manifolds in (n+2)-dimensional manifolds, generalizing the traditional study of knots in the case n=1. The main theme is the application of the author's algebraic theory of surgery to provide a unified treatment of the invariants of codimension 2 embeddings, generalizing the Alexander polynomials and Seifert forms of classical knot theory. Many results in the research literature are thus brought into a single framework, and new results are obtained. The treatment is particularly effective in dealing with open books, which are manifolds with codimension 2 submanifolds such that the complement fibres over a circle. The book concludes with an appendix by E. Winkelnkemper on the history of open books. ... Read more

Customer Reviews (1)

A good introduction to the early work
The finding of invariants for knots has been a major unsolved problem in mathematics for over 125 years. The difficulty of the problem is attested to by the paucity of results over decades of time in the early 20th century. Then in the 1980's Vaughn Jones discovered some invariants that are related to ideas in physics, namely the theory of integrable models in statistical mechanics. This book is a collection of articles discussing the Jones work and other approaches that relate knot theory and statistical mechanics, written a few years after his discovery. My review will be confined to the articles which I read in detail.

An article by Vaughn Jones begins the book and discusses the connection between subfactors of von Neumann algebras and statistical mechanics. These von Neumann algebras occur as algebras of transfer matrices in statistical mechanics. These transfer matrices satisfy algebraic relations that are essentially the same as those appearing in special types of von Neumann algebras. The author makes it a point to discuss in detail the relevant constructions of von Neumann algebras, believing this has not been done in the literature. The von Neumann algebras related to the transfer matrices are particular types of II(1) and III factors, which the author constructs using Bratteli diagrams and the Gelfand-Naimark-Segal construction.

The article by Louis Kauffman discusses polynomial invariants of knots and the Yang-Baxter factorization equation. Polynomial invariants based on the Yang-Baxter equations are one-variable polynomials. The author points out that it is unknown whether two-variable invariants can be extracted from the Yang-Baxter equations, but points out how to construct these using skein models.

The article by Michio Jimbo is an introduction to the Yang-Baxter equation with emphasis on the role of quantum groups. The solutions of the Yang-Baxter equation are discussed in the light of the work of A. Belavin and V.G. Drinfeld in the context of simple Lie algebras. The author shows in this case that the solutions are either elliptic, trigonometric, or rational functions. This is followed by a discussion of how to "quantize" this situation, which leads to the theory of quantum groups, a field that has grown considerably since this article was written. The author discusses a particular example of a quantum group, called the universal enveloping algebra, and studies its representations and the Drinfeld universal R matrix. The "classical" Yang-Baxter r-matrix is then the classical limit of this R-matrix. The author shows how to obtain higher representations by using an analog of the technique of constructing irreducible representations of Lie algebras by forming tensor products of fundamental representations and decomposing them. This technique is known as the fusion procedure here and elsewhere in the literature.

The article by Toshitake Kohno is a review article on representations of the braid group with respect to the Yang-Baxter equation for face models in statistical mechanics. The representations of the braid group appear explicitly as the monodromy of integrable connections defined for any simple Lie algebra and its irreducible representation. Interestingly, the connections describe n-point functions in a conformal field theory on the Riemann sphere with gauge symmetry. The author begins with a finite-dimensional complex simple Lie algebra and its irreducible representation. Selecting an orthonormal basis of this Lie algebra with respect to the Cartan-Killing form, the author constructs certain matrices in the endomorphisms of the n-fold tensor product of the representation. These matrices satisfy certain relations that are a special case of the Yang-Baxter equation. A connection defined using these matrices and a complex parameter ranging over a set consisting complex n-vectors with unequal coordinates is shown to be integrable using these relations. The fundamental group of the complex parameter set is the 'pure braid group with n strings' and the (quadratic) relations are viewed as an infinitesimal version of the defining relations of the pure braid group. By taking the quotient of the parameter set with the symmetric group one obtains the braid group, the representations of which are consequently obtained using the monodromy of this connection. Another representation is derived from the quantized universal enveloping algebra of the Lie algebra.

By far the most interesting article, and the one least rigorous mathematically, is the one by Edward Witten on quantum field theory and the Jones polynomial. The author shows that a Yang-Mills theory in 2 + 1 dimensions consisting of merely the Chern-Simons terms is exactly soluble and can be used to give the Jones polynomial a three-dimensional interpretation, which was highly desired at the time of writing. He also shows that the Jones polynomial can be generalized from the 3-sphere to arbitrary 3-manifolds, and gives invariants for these manifolds, which can be computed from a surgery presentation. The author's constructions are fascinating, particularly from a physics standpoint, but mathematically they are very suspect, since they are dependent on the notion of a path integral. The latter, despite decades of concentrated effort, has defied a mathematically rigorous formulation. The results in the article have thus been classified as "physical mathematics", and therefore conjectural and tentative from a purely mathematical standpoint. This is a fair classification, and it motivated other mathematicians to find alternative formulations that are well-defined mathematically. Indeed, this article has resulted in an explosion of research on both knot invariants and invariants for 3-manifolds, some of which has remain tied to quantum field theory, and some making a concentrated effort to remove these invariants from their dependence on it. ... Read more

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This book sets up a discrete universe with minimum and maximumdimensions. Singularity is rejected.

Entropic Spacetime Theory divides the universe into a kinetic systemand an entropic spacetime. The kinetic system is what our presentphysics is all about; it deals with radiation (vector bosons) and massparticles (fermions). Relativity and quantum mechanics deal almostentirely in the kinetic system.

The entropic spacetime (EST) defines space; in this theory there is novacuum - EST is space. Made up of energy and dipole charges, itsvalues can be converted into length and time.

The theory offers a new description of space, a new cosmology, namesspace as the original creator of all new matter and radiation. ... Read more

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An exposition of building blocks for deformation theory of braided monoidal categories, giving rise to sequences of Vasseliv invariants of framed links, clarifying the interrelations between them. ... Read more

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Braid theory and knot theory are related via two famous results due to Alexander and Markov. Alexander's theorem states that any knot or link can be put into braid form. Markov's theorem gives necessary and sufficient conditions to conclude that two braids represent the same knot or link. Thus, one can use braid theory to study knot theory and vice versa. In this book, the author generalizes braid theory to dimension four. He develops the theory of surface braids and applies it to study surface links. In particular, the generalized Alexander and Markov theorems in dimension four are given. This book is the first to contain a complete proof of the generalized Markov theorem. Surface links are studied via the motion picture method, and some important techniques of this method are studied. For surface braids, various methods to describe them are introduced and developed: the motion picture method, the chart description, the braid monodromy, and the braid system. These tools are fundamental to understanding and computing invariants of surface braids and surface links. Included is a table of knotted surfaces with a computation of Alexander polynomials. Braid techniques are extended to represent link homotopy classes. The book is geared toward a wide audience, from graduate students to specialists. It would make a suitable text for a graduate course and a valuable resource for researchers. ... Read more

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The present volume is an updated version of the book edited byC N Yang and M L Ge on the topics of braid groups and knot theory,which are related to statistical mechanics. This book is based on the1989 volume but has new material included and new contributors. ... Read more

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The lectures in these proceedings discuss topics in statistical mechanics, the geometric and algebraic approaches to q-deformation theories, 2-dimensional gravity and related problems of mathematical physics, including Vassiliev invariants and the Jones polynomials, R-matrix with Z-symmetry, reflection equations and quantum algebra, W-geometry, braid linear algebra, holomorphic q-difference systems and q-Poincare algebra. ... Read more

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This book presents a remarkable application of graph theory to knot theory. In knot theory, there are a number of easily defined geometric invariants that are extremely difficult to compute; the braid index of a knot or link is one example. The authors evaluate the braid index for many knots and links using the generalized Jones polynomial and the index of a graph, a new invariant introduced here. This invariant, which is determined algorithmically, is likely to be of particular interest to computer scientists. ... Read more

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Knot theory is a rapidly developing field of research with many applications not only for mathematics. the present volume, written by a well-known specialist, gives a complete survey of knot theory from its very beginnings to today's most recent research results. The topics include Alexander polynomials, Jones type polynomials, and Vassiliev invariants.

With its appendix containing many useful tables and an extended list of reference with over 3500 entries it is an indispensible book for everyone concerned with knot theory.

The book can serve as an introduction to the field for advanced undergraduate and graduate students. Also researchers working in outside areas such as theoretical physics or molecular biology will benefit from this thorough study which is complemented by many exercises and examples. ... Read more

Customer Reviews (1)

Excellent reference book on knot theory
Just excellent reference on knot theory!

Not an easy read for someone who has no prior knowledge of knot theory. If you have just started studying knot theory or want to study, don't buy it unless you find it at a very low price (like I did :) or you have plans for years of knot studies.

If you want a book with references on many aspects of knot theory, then this is a very good book.

Ypercube ... Read more

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LinKnot Knot Theory by Computer provides a unique view of selected topics in knot theory suitable for students, research mathematicians, and readers with backgrounds in other exact sciences, including chemistry, molecular biology and physics. The book covers basic notions in knot theory, as well as new methods for handling open problems such as unknotting number, braid family representatives, invertibility, amphicheirality, undetectability, non-algebraic tangles, polyhedral links, and (2,2)-moves.Hands-on computations using Mathematica or the webMathematica package LinKnot and beautiful illustrations facilitate better learning and understanding. LinKnot is also a powerful research tool for experimental mathematics implementation of Caudron's ideas. The use of Conway notation enables experimenting with large families of knots and links. Conjectures discussed in the book are explained at length. The beauty, universality and diversity of knot theory is illuminated through various non-standard applications: mirror curves, fullerens, self-referential systems, and KL automata. ... Read more

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Topics in Knot Theory is a state of the art volume whichpresents surveys of the field by the most famous knot theorists in theworld. It also includes the most recent research work by graduate andpostgraduate students. The new ideas presented cover racks,imitations, welded braids, wild braids, surgery, computer calculationsand plottings, presentations of knot groups and representations ofknot and link groups in permutation groups, the complex plane and/orgroups of motions.
For mathematicians, graduate students and scientists interested inknot theory. ... Read more

Customer Reviews (1)

A good reference
This book is a collection of articles that are representative of the many exciting developments in knot theory that were occurring at the time of publication. Many of these developments have been extended and generalized since then, such as the theory of Vassiliev invariants, and thus the articles could be viewed as an introduction to this research. Hence it could still serve as a reference to the mathematical theory of knots and it relation to physics, via statistical mechanics and quantum field theory. I did not read all of the articles, so only a few of the ones I did will be reviewed here.

The article by Vaughan Jones on polynomial invariants for knots via von Neumann algebras begins the collection and was definitely the tone-setting one of the time, due to the new invariants of knots discovered by Jones. The article discusses how to construct a polynomial invariant for tame oriented links using certain representations of the braid group. By using Markov's theorem and a trace on a type II(1) von Neumann algebra, the author shows that the invariant depends only on the closed braid. The von Neumann algebra is generated by an identity and a collection of projections, which satisfy certain types of relations. These relations involve a complex parameter, and when this parameter satisfies certain conditions there exists a trace on the von Neumann algebra which in turn satisfy a collection of relations. The relations on the projections and the trace determine the structure of the von Neumann algebra up to *-isomorphism. That the projection relations are similar to Artin's presentation of the braid group was what Jones and others to develop invariants of links and knots based on this trace. In another article Jones then obtains a polynomial invariant in two variables for oriented links that uses a trace on Hecke algebras "of type A", which was inspired by the connections with von Neumann algebras. His discussion in this article points out the need for a better understanding of the topological interpretation of these invariants. Pointing out that a more in-depth understanding of subfactors of finite index would assist in this topological interpretation, in a later article Jones outlines in more detail what is known for subfactors of finite index. The index, as defined by Jones, measures the size of a subfactor in a II(1) factor. In addition, Hans Wenzl discusses Hecke algebras of type A and subfactors, and shows how to compute the Jones index using AF algebras.

The most provocative article in the book, and one not rigorous from a mathematical standpoint, is the article by Edward Witten on the quantum field theory and the Jones polynomial. The connection between these two seemingly disparate fields caused great excitiment in both the physics and mathematics communities, in spite of the fact that these results are unjustified mathematically, due to their reliance on path integrals. Witten was motivated in this article to find a three-dimensional interpretation of the Jones polynomial, which he does so via Yang-Mills theory in three dimensions. However, the Yang-Mills theory which he uses is not the standard one, but instead is based on the purely topological Chern-Simons theory. Witten considers the quantum field theory defined by the Chern-Simons theory and uses its gauge fields to define gauge-invariant observables. Because of the side-constraint of general covariance, these observables are chosen to be Wilson lines, which are independent of the metric. In an oriented three manifold Witten then considers oriented and non-intersecting knots and assigns a representation to each knots. Using the Chern-Simons three form Witten computes the path integral of the Wilsonobservables, and then proposes that these quantities are 3-dimensional interpretations of the Jones invariant. Witten first proves that the Chern-Simon form gives a meaningful quantum theory, i.e. that it is free from anomalies, and he justifies this by reducing the Chern-Simons invariant to a ratio of determinants, and then showing the absolute value of this ratio is the Ray-Singer analytic torsion. Witten then considers the calculation of the phase of the ratio, and then via the canonical quantization of the theory, shows how to obtain the desired knot invariants. ... Read more

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This book brings together twenty essays on diverse topics inthe history and science of knots. It is divided into five parts, whichdeal respectively with knots in prehistory and antiquity, non-Europeantraditions, working knots, the developing science of knots, anddecorative and other aspects of knots.

Its authors include archaeologists who write on knots found in digs ofancient sites (one describes the knots used by the recently discoveredIce Man); practical knotters who have studied the history and uses ofknots at sea, for fishing and for various life support activities; ahistorian of lace; a computer scientist writing on computerclassification of doilies; and mathematicians who describe the historyof knot theories from the eighteenth century to the present day.

In view of the explosion of mathematical theories of knots in the pastdecade, with consequential new and important scientific applications,this book is timely in setting down a brief, fragmentary history ofmankind's oldest and most useful technical and decorative device - theknot. ... Read more

Customer Reviews (2)

This is not a knot book
If you are looking for a book to teach you how to tie knots, you need to consider Ashley's Knot book. If you want to know the history behind the different types of knots, you definitely need this book. There are diagrams that will help you visualize the knots being discussed, but they generally aren't explicite enough to be of help in learning how to tie the knots unless you already have an idea of how its done. The true focus of this book is in the use of the different types of knots and archelogical data pertaining to it. Overall, its a great book.

KNOTTY KNOWLEDGE
This is a beautiful book, physically & intellectually.The book covers a variety of aspects of "knot", from the mathematical to the practical, including decorative knotting.It is not a guide to knot tying, nor a treatise on topology/knot theory.Rather, here is a collection of articles that shed light on "knot", from authors many of whom belong to the International Guild of Knot Tyers [sic] sharing their love of aspects of "knot".

Admittedly, this is a book for those who love knots, rather than for those who only care to learn a few things about how to tie a few knots; similarly, those who are interested in topology will perhaps not see the couple of related chapters/articles sufficient to justify the cost--they're of an overview nature. ... Read more