One Cocycles And Knot Invariants
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One Cocycles And Knot Invariants
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Author : Thomas Fiedler
language : en
Publisher: World Scientific
Release Date : 2023-01-04
One Cocycles And Knot Invariants written by Thomas Fiedler and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-01-04 with Mathematics categories.
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.
Polynomial One Cocycles For Knots And Closed Braids
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Author : Fiedler Thomas
language : en
Publisher: World Scientific
Release Date : 2019-08-27
Polynomial One Cocycles For Knots And Closed Braids written by Fiedler Thomas and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-08-27 with Mathematics categories.
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.
Surfaces In 4 Space
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Author : Scott Carter
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-06-29
Surfaces In 4 Space written by Scott Carter and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-06-29 with Mathematics categories.
Surfaces in 4-Space, written by leading specialists in the field, discusses knotted surfaces in 4-dimensional space and surveys many of the known results in the area. Results on knotted surface diagrams, constructions of knotted surfaces, classically defined invariants, and new invariants defined via quandle homology theory are presented. The last chapter comprises many recent results, and techniques for computation are presented. New tables of quandles with a few elements and the homology groups thereof are included. This book contains many new illustrations of knotted surface diagrams. The reader of the book will become intimately aware of the subtleties in going from the classical case of knotted circles in 3-space to this higher dimensional case. As a survey, the book is a guide book to the extensive literature on knotted surfaces and will become a useful reference for graduate students and researchers in mathematics and physics.
Seeing Four Dimensional Space And Beyond Using Knots
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Author : Eiji Ogasa
language : en
Publisher: World Scientific
Release Date : 2023-07-21
Seeing Four Dimensional Space And Beyond Using Knots written by Eiji Ogasa and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-07-21 with Mathematics categories.
According to string theory, our universe exists in a 10- or 11-dimensional space. However, the idea the space beyond 3 dimensions seems hard to grasp for beginners. This book presents a way to understand four-dimensional space and beyond: with knots! Beginners can see high dimensional space although they have not seen it.With visual illustrations, we present the manipulation of figures in high dimensional space, examples of which are high dimensional knots and n-spheres embedded in the (n+2)-sphere, and generalize results on relations between local moves and knot invariants into high dimensional space.Local moves on knots, circles embedded in the 3-space, are very important to research in knot theory. It is well known that crossing changes are connected with the Alexander polynomial, the Jones polynomial, HOMFLYPT polynomial, Khovanov homology, Floer homology, Khovanov homotopy type, etc. We show several results on relations between local moves on high dimensional knots and their invariants.The following related topics are also introduced: projections of knots, knot products, slice knots and slice links, an open question: can the Jones polynomial be defined for links in all 3-manifolds? and Khovanov-Lipshitz-Sarkar stable homotopy type. Slice knots exist in the 3-space but are much related to the 4-dimensional space. The slice problem is connected with many exciting topics: Khovanov homology, Khovanv-Lipshits-Sarkar stable homotopy type, gauge theory, Floer homology, etc. Among them, the Khovanov-Lipshitz-Sarkar stable homotopy type is one of the exciting new areas; it is defined for links in the 3-sphere, but it is a high dimensional CW complex in general.Much of the book will be accessible to freshmen and sophomores with some basic knowledge of topology.
Lecture Notes On Knot Invariants
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Author : Weiping Li
language : en
Publisher: World Scientific
Release Date : 2015-08-26
Lecture Notes On Knot Invariants written by Weiping Li and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-08-26 with Mathematics categories.
The volume is focused on the basic calculation skills of various knot invariants defined from topology and geometry. It presents the detailed Hecke algebra and braid representation to illustrate the original Jones polynomial (rather than the algebraic formal definition many other books and research articles use) and provides self-contained proofs of the Tait conjecture (one of the big achievements from the Jones invariant). It also presents explicit computations to the Casson–Lin invariant via braid representations. With the approach of an explicit computational point of view on knot invariants, this user-friendly volume will benefit readers to easily understand low-dimensional topology from examples and computations, rather than only knowing terminologies and theorems. Contents:Basic Knots, Links and Their EquivalencesBraids and LinksKnot and Link InvariantsJones PolynomialsCasson Type Invariants Readership: Undergraduate and graduate students interested in learning topology and low dimensional topology. Key Features:Applies a computational approach to understand knot invariants with geometric meaningsProvides a complete proof of Tait's conjectures from an original Jones polynomial definitionGives recent new knot invariants from the approach of algebraic geometry (characteristic variety)Readers will get a hands-on approach to the topological concepts and various invariant, instead of just knowing more fancy wordsKeywords:Knot Classifications;Tait Conjectures;Reidemeister Moves;Characterization of Braid Representation;Unknotting Number;Bridge Number;Linking Number;Crossing Number;Wirtinger Presentation;Magnus Representation;Twisted Alexander Polynomial;Hecke Algebra;Ocneanu Trace;Jones Polynomial;Kauffman Bracket;Casson Type Invariant
Scientific Legacy Of Professor Zbigniew Oziewicz Selected Papers From The International Conference Applied Category Theory Graph Operad Logic
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Author : Hilda Maria Colin Garcia
language : en
Publisher: World Scientific
Release Date : 2023-09-27
Scientific Legacy Of Professor Zbigniew Oziewicz Selected Papers From The International Conference Applied Category Theory Graph Operad Logic written by Hilda Maria Colin Garcia and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-09-27 with Mathematics categories.
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.
Introductory Lectures On Knot Theory
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Author : Louis H. Kauffman
language : en
Publisher: World Scientific
Release Date : 2012
Introductory Lectures On Knot Theory written by Louis H. Kauffman and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012 with Mathematics categories.
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.
Introduction To Vassiliev Knot Invariants
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Author : S. Chmutov
language : en
Publisher: Cambridge University Press
Release Date : 2012-05-24
Introduction To Vassiliev Knot Invariants written by S. Chmutov and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-05-24 with Mathematics categories.
A detailed exposition of the theory with an emphasis on its combinatorial aspects.
Topology And Geometry Of Manifolds
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Author : Ga.) Georgia International Topology Conference (2001 : Athens
language : en
Publisher: American Mathematical Soc.
Release Date : 2003
Topology And Geometry Of Manifolds written by Ga.) Georgia International Topology Conference (2001 : Athens and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003 with Differential topology categories.
Since 1961, the Georgia Topology Conference has been held every eight years to discuss the newest developments in topology. The goals of the conference are to disseminate new and important results and to encourage interaction among topologists who are in different stages of their careers. Invited speakers are encouraged to aim their talks to a broad audience, and several talks are organized to introduce graduate students to topics of current interest. Each conference results in high-quality surveys, new research, and lists of unsolved problems, some of which are then formally published. Continuing in this 40-year tradition, the AMS presents this volume of articles and problem lists from the 2001 conference. Topics covered include symplectic and contact topology, foliations and laminations, and invariants of manifolds and knots. Articles of particular interest include John Etnyre's, ``Introductory Lectures on Contact Geometry'', which is a beautiful expository paper that explains the background and setting for many of the other papers. This is an excellent introduction to the subject for graduate students in neighboring fields. Etnyre and Lenhard Ng's, ``Problems in Low-Dimensional Contact Topology'' and Danny Calegari's extensive paper,``Problems in Foliations and Laminations of 3-Manifolds'' are carefully selected problems in keeping with the tradition of the conference. They were compiled by Etnyre and Ng and by Calegari with the input of many who were present. This book provides material of current interest to graduate students and research mathematicians interested in the geometry and topology of manifolds.
Knots Low Dimensional Topology And Applications
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Author : Colin C. Adams
language : en
Publisher: Springer
Release Date : 2019-06-26
Knots Low Dimensional Topology And Applications written by Colin C. Adams and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-06-26 with Mathematics categories.
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.