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Grid Homology for Knots and Links

Author: Peter S. Ozsvath

Publisher: American Mathematical Soc.

Published: 2017-01-19

Total Pages: 410

ISBN-13: 1470434423

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

Author: Peter S. Ozsváth

Publisher: American Mathematical Soc.

Published: 2015-12-04

Total Pages: 410

ISBN-13: 1470417375

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Bordered Heegaard Floer Homology

Author: Robert Lipshitz

Publisher: American Mathematical Soc.

Published: 2018-08-09

Total Pages: 279

ISBN-13: 1470428881

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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.

The Mathematics of Knots

Author: Markus Banagl

Publisher: Springer Science & Business Media

Published: 2010-11-25

Total Pages: 363

ISBN-13: 3642156371

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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.

Knots and Links

Author: Peter R. Cromwell

Publisher: Cambridge University Press

Published: 2004-10-14

Total Pages: 356

ISBN-13: 9780521548311

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

Encyclopedia of Knot Theory

Author: Colin Adams

Publisher: CRC Press

Published: 2021-02-10

Total Pages: 1048

ISBN-13: 100022242X

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"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 Introduction to Knot Theory

Author: W.B.Raymond Lickorish

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 213

ISBN-13: 146120691X

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A selection of topics which graduate students have found to be a successful introduction to the field, employing three distinct techniques: geometric topology manoeuvres, combinatorics, and algebraic topology. Each topic is developed until significant results are achieved and each chapter ends with exercises and brief accounts of the latest research. What may reasonably be referred to as knot theory has expanded enormously over the last decade and, while the author describes important discoveries throughout the twentieth century, the latest discoveries such as quantum invariants of 3-manifolds as well as generalisations and applications of the Jones polynomial are also included, presented in an easily intelligible style. Readers are assumed to have knowledge of the basic ideas of the fundamental group and simple homology theory, although explanations throughout the text are numerous and well-done. Written by an internationally known expert in the field, this will appeal to graduate students, mathematicians and physicists with a mathematical background wishing to gain new insights in this area.

Knots, Low-Dimensional Topology and Applications

Author: Colin C. Adams

Publisher: Springer

Published: 2019-06-26

Total Pages: 476

ISBN-13: 3030160319

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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.

Sergei Gukov, Mikhail Khovanov, and Johannes Walcher

Author: Sergei Gukov:

Publisher: American Mathematical Soc.

Published: 2016-12-23

Total Pages: 177

ISBN-13: 1470414597

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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.

Seeing Four-dimensional Space And Beyond: Using Knots!

Author: Eiji Ogasa

Publisher: World Scientific

Published: 2023-07-21

Total Pages: 173

ISBN-13: 9811275165

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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.