Knots In Hellas 98 Proceedings Of The International Conference On Knot Theory And Its Ramifications

Author : V. F. R. Jones
Publisher : World Scientific
Page : 588 pages
File Size : 53,6 Mb
Release : 2000
Category : Mathematics
ISBN : 9812792678

Get Book

Knots in Hellas '98 - Proceedings of the International Conference on Knot Theory and Its Ramifications by V. F. R. Jones Pdf

There have been exciting developments in the area of knot theory in recent years. They include Thurston's work on geometric structures on 3-manifolds (e.g. knot complements), Gordon–Luecke work on surgeries on knots, Jones' work on invariants of links in S3, and advances in the theory of invariants of 3-manifolds based on Jones- and Vassiliev-type invariants of links. Jones ideas and Thurston's idea are connected by the following path: hyperbolic structures, PSL(2, C) representations, character varieties, quantization of the coordinate ring of the variety to skein modules (i.e. Kauffman, bracket skein module), and finally quantum invariants of 3-manifolds. This proceedings volume covers all those exciting topics.

Author : C. McA. Gordon
Publisher : World Scientific
Page : 580 pages
File Size : 46,9 Mb
Release : 2000
Category : Mathematics
ISBN : 9789810243401

Get Book

Knots in Hellas '98 by C. McA. Gordon Pdf

There have been exciting developments in the area of knot theory in recent years. They include Thurston's work on geometric structures on 3-manifolds (e.g. knot complements), Gordon?Luecke work on surgeries on knots, Jones' work on invariants of links in S3, and advances in the theory of invariants of 3-manifolds based on Jones- and Vassiliev-type invariants of links. Jones ideas and Thurston's idea are connected by the following path: hyperbolic structures, PSL(2, C) representations, character varieties, quantization of the coordinate ring of the variety to skein modules (i.e. Kauffman, bracket skein module), and finally quantum invariants of 3-manifolds. This proceedings volume covers all those exciting topics.

Author : Anonim
Publisher : Unknown
Page : 128 pages
File Size : 40,5 Mb
Release : 2024-06-12
Category : Electronic
ISBN : 9789814474030

Get Book

Linknot by Anonim Pdf

Author : Colin Adams,Erica Flapan,Allison Henrich,Louis H. Kauffman,Lewis D. Ludwig,Sam Nelson
Publisher : CRC Press
Page : 1048 pages
File Size : 52,9 Mb
Release : 2021-02-10
Category : Mathematics
ISBN : 9781000222425

Get Book

Encyclopedia of Knot Theory by Colin Adams,Erica Flapan,Allison Henrich,Louis H. Kauffman,Lewis D. Ludwig,Sam Nelson Pdf

"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

Author : Hilda Maria Colin Garcia,Jose De Jesus Cruz Guzman,Louis H Kauffman,Hanna Makaruk
Publisher : World Scientific
Page : 771 pages
File Size : 49,8 Mb
Release : 2023-09-27
Category : Mathematics
ISBN : 9789811271168

Get Book

Scientific Legacy Of Professor Zbigniew Oziewicz: Selected Papers From The International Conference "Applied Category Theory Graph-operad-logic" by Hilda Maria Colin Garcia,Jose De Jesus Cruz Guzman,Louis H Kauffman,Hanna Makaruk Pdf

Dedicated to the memory of the late Professor Zbigniew Oziewicz from Universidad Nacional Autónoma de México, the book consists of papers on a wide variety of topics related to the work of Professor Oziewicz, which were presented at the special conference on Graph-Operads-Logic (GOL 2021), selected through peer review to promote his scientific legacy.Professor Oziewicz was a great enthusiast and supporter of category theory and its applications in physics, as well as in various areas of mathematics (topology, noncommutative geometry, etc.). In particular, he made significant contributions to the theory of Frobenius algebras, which now are becoming more important due to their connection with topological quantum field theories that are used in mathematical physics and in quantum topology. Professor Oziewicz was a great and very generous teacher, who immersed his students in the beautiful ideas of category theory as well as mathematical physics and computation. It was his idea to start a series of conferences under the title Graphs-Operads-Logic, most of them held in Mexico, with some of them in the USA, which were a great platform to discuss various ideas connected with category theory and its various applications, and to make friends with other scientists. Despite his passing, the GOL 2021 conference is included in this series to pay tribute to his many contributions to diverse areas of science.The book is laid out in twelve main topics where we can find relevant works from distinguished experts.

Author : Louis H. Kauffman
Publisher : World Scientific
Page : 577 pages
File Size : 47,5 Mb
Release : 2012
Category : Mathematics
ISBN : 9789814313001

Get Book

Introductory Lectures on Knot Theory by Louis H. Kauffman Pdf

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.

Author : Jorge Alberto Calvo
Publisher : World Scientific
Page : 640 pages
File Size : 40,9 Mb
Release : 2005
Category : Mathematics
ISBN : 9789812561879

Get Book

Physical and Numerical Models in Knot Theory by Jorge Alberto Calvo Pdf

The physical properties of knotted and linked configurations in space have long been of interest to mathematicians. More recently, these properties have become significant to biologists, physicists, and engineers among others. Their depth of importance and breadth of application are now widely appreciated and valuable progress continues to be made each year.This volume presents several contributions from researchers using computers to study problems that would otherwise be intractable. While computations have long been used to analyze problems, formulate conjectures, and search for special structures in knot theory, increased computational power has made them a staple in many facets of the field. The volume also includes contributions concentrating on models researchers use to understand knotting, linking, and entanglement in physical and biological systems. Topics include properties of knot invariants, knot tabulation, studies of hyperbolic structures, knot energies, the exploration of spaces of knots, knotted umbilical cords, studies of knots in DNA and proteins, and the structure of tight knots. Together, the chapters explore four major themes: physical knot theory, knot theory in the life sciences, computational knot theory, and geometric knot theory.

Author : David N. Yetter
Publisher : World Scientific
Page : 244 pages
File Size : 44,6 Mb
Release : 2001
Category : Mathematics
ISBN : 9812810463

Get Book

Functorial Knot Theory by David N. Yetter Pdf

Almost since the advent of skein-theoretic invariants of knots and links (the Jones, HOMFLY, and Kauffman polynomials), the important role of categories of tangles in the connection between low-dimensional topology and quantum-group theory has been recognized. The rich categorical structures naturally arising from the considerations of cobordisms have suggested functorial views of topological field theory. This book begins with a detailed exposition of the key ideas in the discovery of monoidal categories of tangles as central objects of study in low-dimensional topology. The focus then turns to the deformation theory of monoidal categories and the related deformation theory of monoidal functors, which is a proper generalization of Gerstenhaber''s deformation theory of associative algebras. These serve as the building blocks for a deformation theory of braided monoidal categories which gives rise to sequences of Vassiliev invariants of framed links, and clarify their interrelations. Contents: Knots and Categories: Monoidal Categories, Functors and Natural Transformations; A Digression on Algebras; Knot Polynomials; Smooth Tangles and PL Tangles; A Little Enriched Category Theory; Deformations: Deformation Complexes of Semigroupal Categories and Functors; First Order Deformations; Units; Extrinsic Deformations of Monoidal Categories; Categorical Deformations as Proper Generalizations of Classical Notions; and other papers. Readership: Mathematicians and theoretical physicists.

Author : Tomotada Ohtsuki
Publisher : World Scientific
Page : 508 pages
File Size : 40,5 Mb
Release : 2001-12-21
Category : Mathematics
ISBN : 9789814490719

Get Book

Quantum Invariants by Tomotada Ohtsuki Pdf

This book provides an extensive and self-contained presentation of quantum and related invariants of knots and 3-manifolds. Polynomial invariants of knots, such as the Jones and Alexander polynomials, are constructed as quantum invariants, i.e. invariants derived from representations of quantum groups and from the monodromy of solutions to the Knizhnik-Zamolodchikov equation. With the introduction of the Kontsevich invariant and the theory of Vassiliev invariants, the quantum invariants become well-organized. Quantum and perturbative invariants, the LMO invariant, and finite type invariants of 3-manifolds are discussed. The Chern–Simons field theory and the Wess–Zumino–Witten model are described as the physical background of the invariants. Contents: Knots and Polynomial InvariantsBraids and Representations of the Braid GroupsOperator Invariants of Tangles via Sliced DiagramsRibbon Hopf Algebras and Invariants of LinksMonodromy Representations of the Braid Groups Derived from the Knizhnik–Zamolodchikov EquationThe Kontsevich InvariantVassiliev InvariantsQuantum Invariants of 3-ManifoldsPerturbative Invariants of Knots and 3-ManifoldsThe LMO InvariantFinite Type Invariants of Integral Homology 3-Spheres Readership: Researchers, lecturers and graduate students in geometry, topology and mathematical physics. Keywords:Kontsevich Invariant;LMO Invariant;Quantum Groups;Knot;3-Manifold;Quantum Invariant;Vassiliev Invariant;Finite Type Invariant;Chord Diagram;Jacobi Diagram;KZ Equation;Chern-Simons TheoryReviews:“This is a nicely written and useful book: I think that the author has done a great job in explaining quantum invariants of knots and 3-manifolds also on an intuitive and well-motivated, organically growing and not too technical level, at the same time however presenting a lot of material ordered by a clear guiding line.”Mathematics Abstracts “Ohtsuki's book is a very valuable addition to the literature. It surveys the full spectrum of work in the area of quantum invariants … Ohtsuk's book is very readable, for he makes an attempt to present the material in as straightforward a way as possible … the presentation here is very clear and should be easily accessible … this is an excellent book which I would recommend to beginners wanting to learn about quantum invariants and to experts alike.”Mathematical Reviews

Author : Vasiliĭ Olegovich Manturov,Denis Petrovich Ilʹi︠u︡tko
Publisher : World Scientific
Page : 553 pages
File Size : 45,8 Mb
Release : 2012
Category : Mathematics
ISBN : 9789814401128

Get Book

Virtual Knots by Vasiliĭ Olegovich Manturov,Denis Petrovich Ilʹi︠u︡tko Pdf

The book is the first systematic research completely devoted to a comprehensive study of virtual knots and classical knots as its integral part. The book is self-contained and contains up-to-date exposition of the key aspects of virtual (and classical) knot theory. Virtual knots were discovered by Louis Kauffman in 1996. When virtual knot theory arose, it became clear that classical knot theory was a small integral part of a larger theory, and studying properties of virtual knots helped one understand better some aspects of classical knot theory and encouraged the study of further problems. Virtual knot theory finds its applications in classical knot theory. Virtual knot theory occupies an intermediate position between the theory of knots in arbitrary three-manifold and classical knot theory. In this book we present the latest achievements in virtual knot theory including Khovanov homology theory and parity theory due to V O Manturov and graph-link theory due to both authors. By means of parity, one can construct functorial mappings from knots to knots, filtrations on the space of knots, refine many invariants and prove minimality of many series of knot diagrams. Graph-links can be treated as "diagramless knot theory": such "links" have crossings, but they do not have arcs connecting these crossings. It turns out, however, that to graph-links one can extend many methods of classical and virtual knot theories, in particular, the Khovanov homology and the parity theory.

Author : Louis H. Kauffman
Publisher : World Scientific
Page : 865 pages
File Size : 55,9 Mb
Release : 2013
Category : Mathematics
ISBN : 9789814383004

Get Book

Knots and Physics by Louis H. Kauffman Pdf

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.

Author : Fiedler Thomas
Publisher : World Scientific
Page : 260 pages
File Size : 53,5 Mb
Release : 2019-08-27
Category : Mathematics
ISBN : 9789811210310

Get Book

Polynomial One-cocycles For Knots And Closed Braids by Fiedler Thomas Pdf

Traditionally, knot theory deals with diagrams of knots and the search of invariants of diagrams which are invariant under the well known Reidemeister moves. This book goes one step beyond: it gives a method to construct invariants for one parameter famillies of diagrams and which are invariant under 'higher' Reidemeister moves. Luckily, knots in 3-space, often called classical knots, can be transformed into knots in the solid torus without loss of information. It turns out that knots in the solid torus have a particular rich topological moduli space. It contains many 'canonical' loops to which the invariants for one parameter families can be applied, in order to get a new sort of invariants for classical knots.

Author : Jun O'Hara
Publisher : World Scientific
Page : 308 pages
File Size : 49,7 Mb
Release : 2003
Category : Mathematics
ISBN : 9812795308

Get Book

Energy of Knots and Conformal Geometry by Jun O'Hara Pdf

Energy of knots is a theory that was introduced to create a OC canonical configurationOCO of a knot OCo a beautiful knot which represents its knot type. This book introduces several kinds of energies, and studies the problem of whether or not there is a OC canonical configurationOCO of a knot in each knot type. It also considers this problems in the context of conformal geometry. The energies presented in the book are defined geometrically. They measure the complexity of embeddings and have applications to physical knotting and unknotting through numerical experiments. Contents: In Search of the OC Optimal EmbeddingOCO of a Knot: -Energy Functional E (); On E (2); L p Norm Energy with Higher Index; Numerical Experiments; Stereo Pictures of E (2) Minimizers; Energy of Knots in a Riemannian Manifold; Physical Knot Energies; Energy of Knots from a Conformal Geometric Viewpoint: Preparation from Conformal Geometry; The Space of Non-Trivial Spheres of a Knot; The Infinitesimal Cross Ratio; The Conformal Sin Energy E sin (c) Measure of Non-Trivial Spheres; Appendices: Generalization of the Gauss Formula for the Linking Number; The 3-Tuple Map to the Set of Circles in S 3; Conformal Moduli of a Solid Torus; Kirchhoff Elastica; Open Problems and Dreams. Readership: Graduate students and researchers in geometry & topology and numerical & computational mathematics."

Author : Sofia Lambropoulou,Doros Theodorou,Petros Stefaneas,Louis H. Kauffman
Publisher : Springer
Page : 482 pages
File Size : 49,9 Mb
Release : 2017-12-14
Category : Mathematics
ISBN : 9783319681030

Get Book

Algebraic Modeling of Topological and Computational Structures and Applications by Sofia Lambropoulou,Doros Theodorou,Petros Stefaneas,Louis H. Kauffman Pdf

This interdisciplinary book covers a wide range of subjects, from pure mathematics (knots, braids, homotopy theory, number theory) to more applied mathematics (cryptography, algebraic specification of algorithms, dynamical systems) and concrete applications (modeling of polymers and ionic liquids, video, music and medical imaging). The main mathematical focus throughout the book is on algebraic modeling with particular emphasis on braid groups. The research methods include algebraic modeling using topological structures, such as knots, 3-manifolds, classical homotopy groups, and braid groups. The applications address the simulation of polymer chains and ionic liquids, as well as the modeling of natural phenomena via topological surgery. The treatment of computational structures, including finite fields and cryptography, focuses on the development of novel techniques. These techniques can be applied to the design of algebraic specifications for systems modeling and verification. This book is the outcome of a workshop in connection with the research project Thales on Algebraic Modeling of Topological and Computational Structures and Applications, held at the National Technical University of Athens, Greece in July 2015. The reader will benefit from the innovative approaches to tackling difficult questions in topology, applications and interrelated research areas, which largely employ algebraic tools.

Author : Thomas Fiedler
Publisher : World Scientific
Page : 341 pages
File Size : 53,6 Mb
Release : 2023-01-04
Category : Mathematics
ISBN : 9789811263019

Get Book

One-cocycles And Knot Invariants by Thomas Fiedler Pdf

One-Cocycles and Knot Invariants is about classical knots, i.e., smooth oriented knots in 3-space. It introduces discrete combinatorial analysis in knot theory in order to solve a global tetrahedron equation. This new technique is then used to construct combinatorial 1-cocycles in a certain moduli space of knot diagrams. The construction of the moduli space makes use of the meridian and the longitude of the knot. The combinatorial 1-cocycles are therefore lifts of the well-known Conway polynomial of knots, and they can be calculated in polynomial time. The 1-cocycles can distinguish loops consisting of knot diagrams in the moduli space up to homology. They give knot invariants when they are evaluated on canonical loops in the connected components of the moduli space. They are a first candidate for numerical knot invariants which can perhaps distinguish the orientation of knots.