Grid Homology For Knots And Links

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Grid Homology for Knots and Links

Peter S. Ozsvath,András I. Stipsicz,Zoltan Szabo
Grid Homology for Knots and Links Book PDF

Author : Peter S. Ozsvath,András I. Stipsicz,Zoltan Szabo
Publisher : American Mathematical Soc.
Page : 410 pages
File Size : 52,9 Mb
Release : 2017-01-19
Category : Education
ISBN : 9781470434427

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Grid Homology for Knots and Links by Peter S. Ozsvath,András I. Stipsicz,Zoltan Szabo Pdf

Knot theory is a classical area of low-dimensional topology, directly connected with the theory of three-manifolds and smooth four-manifold topology. In recent years, the subject has undergone transformative changes thanks to its connections with a number of other mathematical disciplines, including gauge theory; representation theory and categorification; contact geometry; and the theory of pseudo-holomorphic curves. Starting from the combinatorial point of view on knots using their grid diagrams, this book serves as an introduction to knot theory, specifically as it relates to some of the above developments. After a brief overview of the background material in the subject, the book gives a self-contained treatment of knot Floer homology from the point of view of grid diagrams. Applications include computations of the unknotting number and slice genus of torus knots (asked first in the 1960s and settled in the 1990s), and tools to study variants of knot theory in the presence of a contact structure. Additional topics are presented to prepare readers for further study in holomorphic methods in low-dimensional topology, especially Heegaard Floer homology. The book could serve as a textbook for an advanced undergraduate or part of a graduate course in knot theory. Standard background material is sketched in the text and the appendices.

Grid Homology for Knots and Links

Peter S. Ozsváth,András I. Stipsicz,Zoltán Szabó
Grid Homology for Knots and Links Book PDF

Author : Peter S. Ozsváth,András I. Stipsicz,Zoltán Szabó
Publisher : American Mathematical Soc.
Page : 410 pages
File Size : 52,6 Mb
Release : 2015-12-04
Category : Homology theory
ISBN : 9781470417376

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Grid Homology for Knots and Links by Peter S. Ozsváth,András I. Stipsicz,Zoltán Szabó Pdf

Knot theory is a classical area of low-dimensional topology, directly connected with the theory of three-manifolds and smooth four-manifold topology. In recent years, the subject has undergone transformative changes thanks to its connections with a number of other mathematical disciplines, including gauge theory; representation theory and categorification; contact geometry; and the theory of pseudo-holomorphic curves. Starting from the combinatorial point of view on knots using their grid diagrams, this book serves as an introduction to knot theory, specifically as it relates to some of the above developments. After a brief overview of the background material in the subject, the book gives a self-contained treatment of knot Floer homology from the point of view of grid diagrams. Applications include computations of the unknotting number and slice genus of torus knots (asked first in the 1960s and settled in the 1990s), and tools to study variants of knot theory in the presence of a contact structure. Additional topics are presented to prepare readers for further study in holomorphic methods in low-dimensional topology, especially Heegaard Floer homology. The book could serve as a textbook for an advanced undergraduate or part of a graduate course in knot theory. Standard background material is sketched in the text and the appendices.

The Mathematics of Knots

Markus Banagl,Denis Vogel
The Mathematics of Knots Book PDF

Author : Markus Banagl,Denis Vogel
Publisher : Springer Science & Business Media
Page : 363 pages
File Size : 53,7 Mb
Release : 2010-11-25
Category : Mathematics
ISBN : 9783642156373

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The Mathematics of Knots by Markus Banagl,Denis Vogel Pdf

The present volume grew out of the Heidelberg Knot Theory Semester, organized by the editors in winter 2008/09 at Heidelberg University. The contributed papers bring the reader up to date on the currently most actively pursued areas of mathematical knot theory and its applications in mathematical physics and cell biology. Both original research and survey articles are presented; numerous illustrations support the text. The book will be of great interest to researchers in topology, geometry, and mathematical physics, graduate students specializing in knot theory, and cell biologists interested in the topology of DNA strands.

Encyclopedia of Knot Theory

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

Author : Colin Adams,Erica Flapan,Allison Henrich,Louis H. Kauffman,Lewis D. Ludwig,Sam Nelson
Publisher : CRC Press
Page : 1048 pages
File Size : 52,8 Mb
Release : 2021-02-10
Category : Mathematics
ISBN : 9781000222425

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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

Knots and Links

Peter R. Cromwell
Knots and Links Book PDF

Author : Peter R. Cromwell
Publisher : Cambridge University Press
Page : 356 pages
File Size : 49,6 Mb
Release : 2004-10-14
Category : Mathematics
ISBN : 0521548314

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Knots and Links by Peter R. Cromwell Pdf

A richly illustrated 2004 textbook on knot theory; minimal prerequisites but modern in style and content.

Seeing Four-dimensional Space And Beyond: Using Knots!

Eiji Ogasa
Seeing Four dimensional Space And Beyond Using Knots Book PDF

Author : Eiji Ogasa
Publisher : World Scientific
Page : 173 pages
File Size : 44,7 Mb
Release : 2023-07-21
Category : Mathematics
ISBN : 9789811275166

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Seeing Four-dimensional Space And Beyond: Using Knots! by Eiji Ogasa Pdf

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.

Quantum Field Theory and Manifold Invariants

Daniel S. Freed,Sergei Gukov,Ciprian Manolescu,Constantin Teleman,Ulrike Tillmann
Quantum Field Theory and Manifold Invariants Book PDF

Author : Daniel S. Freed,Sergei Gukov,Ciprian Manolescu,Constantin Teleman,Ulrike Tillmann
Publisher : American Mathematical Society, IAS/Park City Mathematics Institute
Page : 476 pages
File Size : 44,9 Mb
Release : 2021-12-02
Category : Mathematics
ISBN : 9781470461232

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Quantum Field Theory and Manifold Invariants by Daniel S. Freed,Sergei Gukov,Ciprian Manolescu,Constantin Teleman,Ulrike Tillmann Pdf

This volume contains lectures from the Graduate Summer School “Quantum Field Theory and Manifold Invariants” held at Park City Mathematics Institute 2019. The lectures span topics in topology, global analysis, and physics, and they range from introductory to cutting edge. Topics treated include mathematical gauge theory (anti-self-dual equations, Seiberg-Witten equations, Higgs bundles), classical and categorified knot invariants (Khovanov homology, Heegaard Floer homology), instanton Floer homology, invertible topological field theory, BPS states and spectral networks. This collection presents a rich blend of geometry and topology, with some theoretical physics thrown in as well, and so provides a snapshot of a vibrant and fast-moving field. Graduate students with basic preparation in topology and geometry can use this volume to learn advanced background material before being brought to the frontiers of current developments. Seasoned researchers will also benefit from the systematic presentation of exciting new advances by leaders in their fields.

Bordered Heegaard Floer Homology

Robert Lipshitz,Peter Ozsváth,Dylan P. Thurston
Bordered Heegaard Floer Homology Book PDF

Author : Robert Lipshitz,Peter Ozsváth,Dylan P. Thurston
Publisher : American Mathematical Soc.
Page : 279 pages
File Size : 45,8 Mb
Release : 2018-08-09
Category : Floer homology
ISBN : 9781470428884

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Bordered Heegaard Floer Homology by Robert Lipshitz,Peter Ozsváth,Dylan P. Thurston Pdf

The authors construct Heegaard Floer theory for 3-manifolds with connected boundary. The theory associates to an oriented, parametrized two-manifold a differential graded algebra. For a three-manifold with parametrized boundary, the invariant comes in two different versions, one of which (type D) is a module over the algebra and the other of which (type A) is an A∞ module. Both are well-defined up to chain homotopy equivalence. For a decomposition of a 3-manifold into two pieces, the A∞ tensor product of the type D module of one piece and the type A module from the other piece is ^HF of the glued manifold. As a special case of the construction, the authors specialize to the case of three-manifolds with torus boundary. This case can be used to give another proof of the surgery exact triangle for ^HF. The authors relate the bordered Floer homology of a three-manifold with torus boundary with the knot Floer homology of a filling.

Differential and Low-Dimensional Topology

András Juhász
Differential and Low Dimensional Topology Book PDF

Author : András Juhász
Publisher : Cambridge University Press
Page : 240 pages
File Size : 48,8 Mb
Release : 2023-03-31
Category : Mathematics
ISBN : 9781009220583

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Differential and Low-Dimensional Topology by András Juhász Pdf

The new student in differential and low-dimensional topology is faced with a bewildering array of tools and loosely connected theories. This short book presents the essential parts of each, enabling the reader to become 'literate' in the field and begin research as quickly as possible. The only prerequisite assumed is an undergraduate algebraic topology course. The first half of the text reviews basic notions of differential topology and culminates with the classification of exotic seven-spheres. It then dives into dimension three and knot theory. There then follows an introduction to Heegaard Floer homology, a powerful collection of modern invariants of three- and four-manifolds, and of knots, that has not before appeared in an introductory textbook. The book concludes with a glimpse of four-manifold theory. Students will find it an exhilarating and authoritative guide to a broad swathe of the most important topics in modern topology.

Knots in Hellas '98 - Proceedings of the International Conference on Knot Theory and Its Ramifications

V. F. R. Jones
Knots in Hellas 98 Proceedings of the International Conference on Knot Theory and Its Ramifications Book PDF

Author : V. F. R. Jones
Publisher : World Scientific
Page : 588 pages
File Size : 48,8 Mb
Release : 2000
Category : Mathematics
ISBN : 9812792678

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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.

Sergei Gukov, Mikhail Khovanov, and Johannes Walcher

Sergei Gukov:,Mikhail Khovanov,Johannes Walcher
Sergei Gukov Mikhail Khovanov and Johannes Walcher Book PDF

Author : Sergei Gukov:,Mikhail Khovanov,Johannes Walcher
Publisher : American Mathematical Soc.
Page : 177 pages
File Size : 50,7 Mb
Release : 2016-12-23
Category : Curves
ISBN : 9781470414597

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Sergei Gukov, Mikhail Khovanov, and Johannes Walcher by Sergei Gukov:,Mikhail Khovanov,Johannes Walcher Pdf

Throughout recent history, the theory of knot invariants has been a fascinating melting pot of ideas and scientific cultures, blending mathematics and physics, geometry, topology and algebra, gauge theory, and quantum gravity. The 2013 Séminaire de Mathématiques Supérieures in Montréal presented an opportunity for the next generation of scientists to learn in one place about the various perspectives on knot homology, from the mathematical background to the most recent developments, and provided an access point to the relevant parts of theoretical physics as well. This volume presents a cross-section of topics covered at that summer school and will be a valuable resource for graduate students and researchers wishing to learn about this rapidly growing field.

Knot Theory

Vassily Olegovich Manturov
Knot Theory Book PDF

Author : Vassily Olegovich Manturov
Publisher : CRC Press
Page : 528 pages
File Size : 53,9 Mb
Release : 2018-04-17
Category : Mathematics
ISBN : 9781351359122

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Knot Theory by Vassily Olegovich Manturov Pdf

Over the last fifteen years, the face of knot theory has changed due to various new theories and invariants coming from physics, topology, combinatorics and alge-bra. It suffices to mention the great progress in knot homology theory (Khovanov homology and Ozsvath-Szabo Heegaard-Floer homology), the A-polynomial which give rise to strong invariants of knots and 3-manifolds, in particular, many new unknot detectors. New to this Edition is a discussion of Heegaard-Floer homology theory and A-polynomial of classical links, as well as updates throughout the text. Knot Theory, Second Edition is notable not only for its expert presentation of knot theory’s state of the art but also for its accessibility. It is valuable as a profes-sional reference and will serve equally well as a text for a course on knot theory.

Knots, Low-Dimensional Topology and Applications

Colin C. Adams,Cameron McA. Gordon,Vaughan F.R. Jones,Louis H. Kauffman,Sofia Lambropoulou,Kenneth C. Millett,Jozef H. Przytycki,Renzo Ricca,Radmila Sazdanovic
Knots Low Dimensional Topology and Applications Book PDF

Author : Colin C. Adams,Cameron McA. Gordon,Vaughan F.R. Jones,Louis H. Kauffman,Sofia Lambropoulou,Kenneth C. Millett,Jozef H. Przytycki,Renzo Ricca,Radmila Sazdanovic
Publisher : Springer
Page : 476 pages
File Size : 43,7 Mb
Release : 2019-06-26
Category : Mathematics
ISBN : 9783030160319

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Knots, Low-Dimensional Topology and Applications by Colin C. Adams,Cameron McA. Gordon,Vaughan F.R. Jones,Louis H. Kauffman,Sofia Lambropoulou,Kenneth C. Millett,Jozef H. Przytycki,Renzo Ricca,Radmila Sazdanovic Pdf

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.

Encyclopedia of Knot Theory

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

Author : Colin Adams,Erica Flapan,Allison Henrich,Louis H. Kauffman,Lewis D. Ludwig,Sam Nelson
Publisher : CRC Press
Page : 954 pages
File Size : 42,6 Mb
Release : 2021-02-10
Category : Education
ISBN : 9781000222388

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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

Introductory Lectures on Knot Theory

Louis H. Kauffman
Introductory Lectures on Knot Theory Book PDF

Author : Louis H. Kauffman
Publisher : World Scientific
Page : 578 pages
File Size : 54,9 Mb
Release : 2012
Category : Mathematics
ISBN : 9789814307994

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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.