Louis H. Kauffman

Knots and Physics (Series on Knots and Everything, Vol 1)

This exploration of combinatorics and knot theory is geared toward advanced undergraduates and graduate students. The author, Louis H. Kauffman, is a professor in the Department of Mathematics, Statistics, and Computer Science at the University of Illinois at Chicago. Kauffman draws upon his work as a topologist to illustrate the relationships between knot theory and statistical mechanics, quantum theory, and algebra, as well as the role of knot theory in combinatorics. Featured topics include state, trails, and the clock theorem; state polynomials and the duality conjecture; knots and links; axiomatic link calculations; spanning surfaces; the genus of alternative links; and ribbon knots and the Arf invariant. Key concepts are related in easy-to-remember terms, and numerous helpful diagrams appear throughout the text. The author has provided a new supplement, entitled "Remarks on Formal Knot Theory," as well as his article, "New Invariants in the Theory of Knots," first published in The American Mathematical Monthly, March 1988.

Formal Knot Theory

Yumei Dang

Daniel J. Sandin

Includes an interactive tour of the space of hypercomplex Julia sets and an educational mini-documentary introducing fractals and hypercomplex geometry.

Distance Estimation and Higher Dimensional Fractals

Hypercomplex Iterations

This invaluable book is an introduction to knot and link invariants as generalised amplitudes for a quasi-physical process. The demands of knot theory, coupled with a quantum-statistical framework, create a context that naturally and powerfully includes a extraordinary range of interrelated topics in topology and mathematical physics. The author takes a primarily combinatorial stance toward knot theory and its relations with these subjects. This stance has the advantage of providing direct access to the algebra and to the combinatorial topology, as well as physical ideas. The book is divided into two parts: Part I is a systematic course on knots and physics starting from the ground up, and Part II is a set of lectures on various topics related to Part I. Part II includes topics such as frictional properties of knots, relations with combinatorics, and knots in dynamical systems. In this third edition, a paper by the author entitled "Functional Integration and Vassiliev invariants" has been added. This paper shows how the Kontsevich integral approach to the Vassiliev invariants is directly related to the perturbative expansion of Witten's functional integral. While the book supplies the background, this paper can be read independently as an introduction to quantum field theory and knot invariants and their relation to quantum gravity. As in the second edition, there is a selection of papers by the author at the end of the book. Numerous clarifying remarks have been added to the text.

Knots and Physics

David E. Radford

Fernando José Oliveira Souza

Hopf algebras have proved to be very interesting structures with deep connections to various areas of mathematics, particularly through quantum groups. Indeed, the study of Hopf algebras, their representations, their generalizations, and the categories related to all these objects has an interdisciplinary nature. It finds methods, relationships, motivations and applications throughout algebra, category theory, topology, geometry, quantum field theory, quantum gravity, and also combinatorics, logic, and theoretical computer science. This volume portrays the vitality of contemporary research in Hopf algebras. Altogether, the articles in the volume explore essential aspects of Hopf algebras and some of their best-known generalizations by means of a variety of approaches and perspectives. They make use of quite different techniques that are already consolidated in the area of quantum algebra. This volume demonstrates the diversity and richness of its subject. Most of its papers introduce the reader to their respective contexts and structures through very expository preliminary sections.

AMS Special Session on Hopf Algebras at the Crossroads of Algebra, Category Theory, and Topology, October 23-24, 2004, Evanston, Illinois

Hopf Algebras and Generalizations

On Knots is a journey through the theory of knots, starting from the simplest combinatorial ideas--ideas arising from the representation of weaving patterns. From this beginning, topological invariants are constructed directly: first linking numbers, then the Conway polynomial and skein theory. This paves the way for later discussion of the recently discovered Jones and generalized polynomials. The central chapter, Chapter Six, is a miscellany of topics and recreations. Here the reader will find the quaternions and the belt trick, a devilish rope trick, Alhambra mosaics, Fibonacci trees, the topology of DNA, and the author's geometric interpretation of the generalized Jones Polynomial. Then come branched covering spaces, the Alexander polynomial, signature theorems, the work of Casson and Gordon on slice knots, and a chapter on knots and algebraic singularities.The book concludes with an appendix about generalized polynomials.

On Knots

An introduction to knot and link invariants as generalised amplitudes for a quasi-physical process. 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.

Sofia Lambropoulou

Colin C. Adams

Cameron McA. Gordon

Vaughan F.R. Jones

Kenneth C. Millett

Jozef H. Przytycki

Renzo Ricca

Radmila Sazdanovic

This proceedings volume presents a diverse collection of high-quality, state-of-the-art research and survey articles written by top experts in low-dimensional topology and its applications. The focal topics include the wide range of historical and contemporary invariants of knots and links and related topics such as three- and four-dimensional manifolds, braids, virtual knot theory, quantum invariants, braids, skein modules and knot algebras, link homology, quandles and their homology; hyperbolic knots and geometric structures of three-dimensional manifolds; the mechanism of topological surgery in physical processes, knots in Nature in the sense of physical knots with applications to polymers, DNA enzyme mechanisms, and protein structure and function. The contents is based on contributions presented at the International Conference on Knots, Low-Dimensional Topology and Applications – Knots in Hellas 2016, which was held at the International Olympic Academy in Greece in July 2016. The goal of the international conference was to promote the exchange of methods and ideas across disciplines and generations, from graduate students to senior researchers, and to explore fundamental research problems in the broad fields of knot theory and low-dimensional topology. This book will benefit all researchers who wish to take their research in new directions, to learn about new tools and methods, and to discover relevant and recent literature for future study.

Knots in Hellas, International Olympic Academy, Greece, July 2016

Knots, Low-Dimensional Topology and Applications

More recently, Khovanov introduced link homology as a generalization of the Jones polynomial to homology of chain complexes and Ozsvath and Szabo developed Heegaard-Floer homology, that lifts the Alexander polynomial. These two significantly different theories are closely related and the dependencies are the object of intensive study. These ideas mark the beginning of a new era in knot theory that includes relationships with four-dimensional problems and the creation of new forms of algebraic topology relevant to knot theory. The theory of skein modules is an older development also having its roots in Jones discovery. Another significant and related development is the theory of virtual knots originated independently by Kauffman and by Goussarov Polyak and Viro in the '90s. All these topics and their relationships are the subject of the survey papers in this book.

Selected Lectures Presented at the Advanced School and Conference on Knot Theory and Its Applications to Physics and Biology, ICTP, Trieste, Italy, 11 - 29 May 2009

Introductory Lectures on Knot Theory

Lynnclaire Dennis

Jytte Brender McNair

Mereon is an approach to the unification of knowledge that relies on whole systems modelling. It is a scientific framework that charts the sequential, emergent growth process of systems. A dynamic structure, Mereon provides insight and a new approach to General Systems Theory and non-linear science. Mereon evolved through a new approach to polyhedral geometry and topology that is related to the dynamics of the polyhedra. It is related to a large number of systems, physical, mathematical, and philosophical. In linking these systems, Mereon provides access to new relationships among them and combines geometric and process thinking. This book provides the fundamentals of such connections for an ongoing search for order, directionality, and diversity that is found in this unity. It is written in clear language that manages to connect diverse disciplines and in doing so, makes a complex system easily accessible and understandable. It will be of interest to mathematicians, geneticists, and all those interested in researching unity in science and astrobiology. Elaborates on several important aspects of General Systems Theory including nonlinearity. Each chapter is self-contained and explained relative to Mereon, providing references to scientific findings that are congruent with or expanded by Mereon. Offers a new way of modelling that can be applied across the sciences.

Unity, Perspective and Paradox

The Mereon Matrix

Andrzej Stasiak

Vsevolod Katritch

In this book, experts in different fields of mathematics, physics, chemistry and biology present unique forms of knots which satisfy certain preassigned criteria relevant to a given field. They discuss the shapes of knotted magnetic flux lines, the forms of knotted arrangements of bistable chemical systems, the trajectories of knotted solitons, and the shapes of knots which can be tied using the shortest piece of elastic rope with a constant diameter.

Ideal Knots

This volume is a collection of research papers devoted to the study of relationships between knot theory and the foundations of mathematics, physics, chemistry, biology and psychology. Included are reprints of the work of Lord Kelvin (Sir William Thomson) on the 19th century theory of vortex atoms, reprints of modern papers on knotted flux in physics and in fluid dynamics and knotted wormholes in general relativity. It also includes papers on Witten's approach to knots via quantum field theory and applications of this approach to quantum gravity and the Ising model in three dimensions. Other papers discuss the topology of RNA folding in relation to invariants of graphs and Vassiliev invariants, the entanglement structures of polymers, the synthesis of molecular Mobius strips and knotted molecules. The book begins with an article on the applications of knot theory to the foundations of mathematics and ends with an article on topology and visual perception. This volume will be of immense interest to all workers interested in new possibilities in the uses of knots and knot theory.

Knots and Applications

This Special Issue of Cybernetics and Human Knowing contains rare material related to G. Spencer-Brown's book Laws of Form and its contents. In 1973 there was a conference at Big Sur at which Spencer-Brown discussed his calculus with a group of scientists. This was the AUM Conference at Esalen, and the scientists consisted in an assortment of remarkable individuals exploring the cutting edge of human consciousness and culture, including Alan Watts, Ram Dass, John Lilly, Heinz von Foerster, Kurt von Meier, and others. One of the participants, Walter Barney, has written about this conference and has long been a keeper of the transcripts of Spencer-Brown's talks. In this issue we print Barney's transcripts of the conference and an article by Walter Barney and Kurt von Meier reflecting on the AUM conference. The transcripts are a remarkable amalgam of the thinking of Spencer-Brown and the questions and comments of the participants in AUM. The transcripts carry the same lucidity that infuses Laws of Form. The other articles in this issue include a paper on Flagg Resolution by James Flagg and Louis Kauffman, a paper on Paper Computers and the Emergence of Fermions by Louis Kauffman, and a Virtual Logic Column by Louis Kauffman that is a new take on the Barber paradox and the Russell Paradox, based on satire, mirrors, and the key observation of Douglas Harding that no person can (in the absence of mirrors) perceive his or her own head. There is an American Society for Cybernetics Column by Zane Gillespie about the structure of implausibility in music, art, and cybernetics.

Laws of Form: Spencer-Brown at Esalen, 1973

This volume is a collection of articles on themes related to the book Laws of Form by George Spencer-Brown. Laws of Form was first published in 1969 and brings forth a new articulation of the foundations of thought. In Laws of Form we have a mathematical formalism based on one symbol and an approach to the question how the world would appear if a distinction could be drawn. Laws of Form does not answer the question how, given nothing as a beginning, a distinction can, indeed must, inevitably take place. This second question must, in its own structure, be left to each individual thinker. Nevertheless, Laws of Form, beautifully written and content free (form is emptiness, emptiness is form) is the most powerful mathematical text on the edge of nothing that has been produced since Euclid's Elements. These papers are a tribute to Spencer-Brown and his singular achievement.

Commentary and Remembrance for George Spencer-Brown

Laws of Form

Soren Brier

This special double issue of Cybernetics and Human Knowing is comprised of a collection of papers devoted to the cybernetics and mathematics of Charles Sanders Peirce with a special focus on its synergies with George Spencer-Brown's thinking. Peirce was a truly original American philosopher and logician working in the late 1800s and early 1900s; Spencer-Brown is an English polymath, best known as the author of Laws of Form. The contributions reflect the extraordinary richness of Peirce's work and his relevance to present concerns in cybernetics. The similarities in the focus on some of the deep foundational subjects are astonishing, amongst those especially the concept of the void or Firstness and the continuity of mind and matter.

History and Synergies in Cybersemiotics

Peirce and Spencer-Brown

Sam Nelson

Allison Henrich

Lewis D. Ludwig

Colin Adams

Erica Flapan

"Knot theory is a fascinating mathematical subject, with multiple links to theoretical physics. This enyclopedia is filled with valuable information on a rich and fascinating subject." – Ed Witten, Recipient of the Fields Medal "I spent a pleasant afternoon perusing the Encyclopedia of Knot Theory. It’s a comprehensive compilation of clear introductions to both classical and very modern developments in the field. It will be a terrific resource for the accomplished researcher, and will also be an excellent way to lure students, both graduate and undergraduate, into the field." – Abigail Thompson, Distinguished Professor of Mathematics at University of California, Davis Knot theory has proven to be a fascinating area of mathematical research, dating back about 150 years. Encyclopedia of Knot Theory provides short, interconnected articles on a variety of active areas in knot theory, and includes beautiful pictures, deep mathematical connections, and critical applications. Many of the articles in this book are accessible to undergraduates who are working on research or taking an advanced undergraduate course in knot theory. More advanced articles will be useful to graduate students working on a related thesis topic, to researchers in another area of topology who are interested in current results in knot theory, and to scientists who study the topology and geometry of biopolymers. Features Provides material that is useful and accessible to undergraduates, postgraduates, and full-time researchers Topics discussed provide an excellent catalyst for students to explore meaningful research and gain confidence and commitment to pursuing advanced degrees Edited and contributed by top researchers in the field of knot theory

Louis H. Kauffman

About

Books

Join the club