Introductory Lectures On Knot Theory Selected Lectures Presented At The Advanced School And Conference On Knot Theory And Its Applications To Physics And Biology
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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.
New Ideas In Low Dimensional Topology
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Author : Vassily Olegovich Manturov
language : en
Publisher: World Scientific
Release Date : 2015-01-27
New Ideas In Low Dimensional Topology written by Vassily Olegovich Manturov and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-01-27 with Mathematics categories.
This book consists of a selection of articles devoted to new ideas and developments in low dimensional topology. Low dimensions refer to dimensions three and four for the topology of manifolds and their submanifolds. Thus we have papers related to both manifolds and to knotted submanifolds of dimension one in three (classical knot theory) and two in four (surfaces in four dimensional spaces). Some of the work involves virtual knot theory where the knots are abstractions of classical knots but can be represented by knots embedded in surfaces. This leads both to new interactions with classical topology and to new interactions with essential combinatorics.
Lectures In Knot Theory
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Author : Józef H. Przytycki
language : en
Publisher: Springer Nature
Release Date : 2024-03-15
Lectures In Knot Theory written by Józef H. Przytycki and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-03-15 with Mathematics categories.
This text is based on lectures delivered by the first author on various, often nonstandard, parts of knot theory and related subjects. By exploring contemporary topics in knot theory including those that have become mainstream, such as skein modules, Khovanov homology and Gram determinants motivated by knots, this book offers an innovative extension to the existing literature. Each lecture begins with a historical overview of a topic and gives motivation for the development of that subject. Understanding of most of the material in the book requires only a basic knowledge of topology and abstract algebra. The intended audience is beginning and advanced graduate students, advanced undergraduate students, and researchers interested in knot theory and its relations with other disciplines within mathematics, physics, biology, and chemistry. Inclusion of many exercises, open problems, and conjectures enables the reader to enhance their understanding of the subject. The use of this text for the classroom is versatile and depends on the course level and choices made by the instructor. Suggestions for variations in course coverage are included in the Preface. The lecture style and array of topical coverage are hoped to inspire independent research and applications of the methods described in the book to other disciplines of science. An introduction to the topology of 3-dimensional manifolds is included in Appendices A and B. Lastly, Appendix C includes a Table of Knots.
Encyclopedia Of Knot Theory
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Author : Colin Adams
language : en
Publisher: CRC Press
Release Date : 2021-02-10
Encyclopedia Of Knot Theory written by Colin Adams and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-02-10 with Education categories.
"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
An Interactive Introduction To Knot Theory
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Author : Inga Johnson
language : en
Publisher: Courier Dover Publications
Release Date : 2017-01-04
An Interactive Introduction To Knot Theory written by Inga Johnson and has been published by Courier Dover Publications this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-01-04 with Mathematics categories.
Well-written and engaging, this hands-on approach features many exercises to be completed by readers. Topics include knot definition and equivalence, combinatorial and algebraic invariants, unknotting operations, and virtual knots. 2016 edition.
Combinatorial Knot Theory
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Author : Roger A Fenn
language : en
Publisher: World Scientific
Release Date : 2024-11-27
Combinatorial Knot Theory written by Roger A Fenn and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-11-27 with Mathematics categories.
A classic knot is an embedded simple loop in 3-dimensional space. It can be described as a 4-valent planar graph or network in the horizontal plane, with the vertices or crossings corresponding to double points of a projection. At this stage we have the shadow of the knot defined by the projection. We can reconstruct the knot by lifting the crossings into two points in space, one above the other. This information is preserved at the vertices by cutting the arc which appears to go under the over crossing arc. We can then act on this diagram of the knot using the famous Reidemeister moves to mimic the motion of the knot in space. The result is classic combinatorial knot theory. In recent years, many different types of knot theories have been considered where the information stored at the crossings determines how the Reidemeister moves are used, if at all.In this book, we look at all these new theories systematically in a way which any third-year undergraduate mathematics student would understand. This book can form the basis of an undergraduate course or as an entry point for a postgraduate studying topology.
Virtual Knots
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Author : Vasilii Olegovich Manturov
language : en
Publisher: World Scientific
Release Date : 2012
Virtual Knots written by Vasilii Olegovich Manturov 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.
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 OC diagramless knot theoryOCO: such OC linksOCO 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.
Victorian Literature And The Physics Of The Imponderable
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Author : Sarah C Alexander
language : en
Publisher: Routledge
Release Date : 2015-07-28
Victorian Literature And The Physics Of The Imponderable written by Sarah C Alexander and has been published by Routledge this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-07-28 with History categories.
The Victorians were obsessed with the empirical but were frequently frustrated by the sizeable gaps in their understanding of the world around them. This study examines how literature and popular culture adopted the emerging language of physics to explain the unknown or ‘imponderable’.
Log Linear Models Extensions And Applications
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Author : Aleksandr Aravkin
language : en
Publisher: MIT Press
Release Date : 2024-12-03
Log Linear Models Extensions And Applications written by Aleksandr Aravkin and has been published by MIT Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-12-03 with Computers categories.
Advances in training models with log-linear structures, with topics including variable selection, the geometry of neural nets, and applications. Log-linear models play a key role in modern big data and machine learning applications. From simple binary classification models through partition functions, conditional random fields, and neural nets, log-linear structure is closely related to performance in certain applications and influences fitting techniques used to train models. This volume covers recent advances in training models with log-linear structures, covering the underlying geometry, optimization techniques, and multiple applications. The first chapter shows readers the inner workings of machine learning, providing insights into the geometry of log-linear and neural net models. The other chapters range from introductory material to optimization techniques to involved use cases. The book, which grew out of a NIPS workshop, is suitable for graduate students doing research in machine learning, in particular deep learning, variable selection, and applications to speech recognition. The contributors come from academia and industry, allowing readers to view the field from both perspectives. Contributors Aleksandr Aravkin, Avishy Carmi, Guillermo A. Cecchi, Anna Choromanska, Li Deng, Xinwei Deng, Jean Honorio, Tony Jebara, Huijing Jiang, Dimitri Kanevsky, Brian Kingsbury, Fabrice Lambert, Aurélie C. Lozano, Daniel Moskovich, Yuriy S. Polyakov, Bhuvana Ramabhadran, Irina Rish, Dimitris Samaras, Tara N. Sainath, Hagen Soltau, Serge F. Timashev, Ewout van den Berg
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.