Surface-Knots in 4-Space

Surface-Knots in 4-Space

Author: Seiichi Kamada

Publisher: Springer

Published: 2017-03-28

Total Pages: 212

ISBN-13: 9811040915

DOWNLOAD EBOOK

This introductory volume provides the basics of surface-knots and related topics, not only for researchers in these areas but also for graduate students and researchers who are not familiar with the field.Knot theory is one of the most active research fields in modern mathematics. Knots and links are closed curves (one-dimensional manifolds) in Euclidean 3-space, and they are related to braids and 3-manifolds. These notions are generalized into higher dimensions. Surface-knots or surface-links are closed surfaces (two-dimensional manifolds) in Euclidean 4-space, which are related to two-dimensional braids and 4-manifolds. Surface-knot theory treats not only closed surfaces but also surfaces with boundaries in 4-manifolds. For example, knot concordance and knot cobordism, which are also important objects in knot theory, are surfaces in the product space of the 3-sphere and the interval.Included in this book are basics of surface-knots and the related topics of classical knots, the motion picture method, surface diagrams, handle surgeries, ribbon surface-knots, spinning construction, knot concordance and 4-genus, quandles and their homology theory, and two-dimensional braids.


Book Synopsis Surface-Knots in 4-Space by : Seiichi Kamada

Download or read book Surface-Knots in 4-Space written by Seiichi Kamada and published by Springer. This book was released on 2017-03-28 with total page 212 pages. Available in PDF, EPUB and Kindle. Book excerpt: This introductory volume provides the basics of surface-knots and related topics, not only for researchers in these areas but also for graduate students and researchers who are not familiar with the field.Knot theory is one of the most active research fields in modern mathematics. Knots and links are closed curves (one-dimensional manifolds) in Euclidean 3-space, and they are related to braids and 3-manifolds. These notions are generalized into higher dimensions. Surface-knots or surface-links are closed surfaces (two-dimensional manifolds) in Euclidean 4-space, which are related to two-dimensional braids and 4-manifolds. Surface-knot theory treats not only closed surfaces but also surfaces with boundaries in 4-manifolds. For example, knot concordance and knot cobordism, which are also important objects in knot theory, are surfaces in the product space of the 3-sphere and the interval.Included in this book are basics of surface-knots and the related topics of classical knots, the motion picture method, surface diagrams, handle surgeries, ribbon surface-knots, spinning construction, knot concordance and 4-genus, quandles and their homology theory, and two-dimensional braids.

Surfaces in 4-Space

Surfaces in 4-Space

Author: Scott Carter

Publisher: Springer Science & Business Media

Published: 2013-06-29

Total Pages: 220

ISBN-13: 3662101629

DOWNLOAD EBOOK

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.


Book Synopsis Surfaces in 4-Space by : Scott Carter

Download or read book Surfaces in 4-Space written by Scott Carter and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 220 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Knotted Surfaces and Their Diagrams

Knotted Surfaces and Their Diagrams

Author: J. Scott Carter

Publisher: American Mathematical Soc.

Published: 1998

Total Pages: 273

ISBN-13: 0821805932

DOWNLOAD EBOOK

In this text, the authors develop the theory of knotted surfaces in analogy with the classical case of knotted curves in three-dimensional space. Knotted surface diagrams are defined; the theory of Reidemeister moves is developed; and the braid theory of knotted surfaces is


Book Synopsis Knotted Surfaces and Their Diagrams by : J. Scott Carter

Download or read book Knotted Surfaces and Their Diagrams written by J. Scott Carter and published by American Mathematical Soc.. This book was released on 1998 with total page 273 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this text, the authors develop the theory of knotted surfaces in analogy with the classical case of knotted curves in three-dimensional space. Knotted surface diagrams are defined; the theory of Reidemeister moves is developed; and the braid theory of knotted surfaces is

The Knot Book

The Knot Book

Author: Colin Conrad Adams

Publisher: American Mathematical Soc.

Published: 2004

Total Pages: 330

ISBN-13: 0821836781

DOWNLOAD EBOOK

Knots are familiar objects. Yet the mathematical theory of knots quickly leads to deep results in topology and geometry. This work offers an introduction to this theory, starting with our understanding of knots. It presents the applications of knot theory to modern chemistry, biology and physics.


Book Synopsis The Knot Book by : Colin Conrad Adams

Download or read book The Knot Book written by Colin Conrad Adams and published by American Mathematical Soc.. This book was released on 2004 with total page 330 pages. Available in PDF, EPUB and Kindle. Book excerpt: Knots are familiar objects. Yet the mathematical theory of knots quickly leads to deep results in topology and geometry. This work offers an introduction to this theory, starting with our understanding of knots. It presents the applications of knot theory to modern chemistry, biology and physics.

How Surfaces Intersect in Space

How Surfaces Intersect in Space

Author: J Scott Carter

Publisher: World Scientific

Published: 1995-05-11

Total Pages: 338

ISBN-13: 9814501239

DOWNLOAD EBOOK

This marvelous book of pictures illustrates the fundamental concepts of geometric topology in a way that is very friendly to the reader. The first chapter discusses the meaning of surface and space and gives the classification of orientable surfaces. In the second chapter we are introduced to the Möbius band and surfaces that can be constructed from this non-orientable piece of fabric. In chapter 3, we see how curves can fit in surfaces and how surfaces can fit into spaces with these curves on their boundary. Basic applications to knot theory are discussed and four-dimensional space is introduced. In Chapter 4 we learn about some 3-dimensional spaces and surfaces that sit inside them. These surfaces help us imagine the structures of the larger space. Chapter 5 is completely new! It contains recent results of Cromwell, Izumiya and Marar. One of these results is a formula relating the rank of a surface to the number of triple points. The other major result is a collection of examples of surfaces in 3-space that have one triple point and 6 branch points. These are beautiful generalizations of the Steiner Roman surface. Chapter 6 reviews the movie technique for examining surfaces in 4-dimensional space. Various movies of the Klein bottle are presented, and the Carter-Saito movie move theorem is explained. The author shows us how to turn the 2-sphere inside out by means of these movie moves and this illustration alone is well worth the price of the book! In the last chapter higher dimensional spaces are examined from an elementary point of view. This is a guide book to a wide variety of topics. It will be of value to anyone who wants to understand the subject by way of examples. Undergraduates, beginning graduate students, and non-professionals will profit from reading the book and from just looking at the pictures. Contents:Front MatterSurface and SpaceNon-orientable SurfacesCurves and KnotsOther Three Dimensional SpacesRelationshipsSurfaces in 4-DimensionsHigher Dimensional SpacesBack Matter Readership: Undergraduates, graduates and mathematicians. keywords:Moving Surfaces;Surfaces;Triple Point;Branch Points “In this excellent book the author teaches us to see a bit more than it meets our eyes. Without hurry he introduces us to the world of topological images. Step by step the reader learns the beauty of topological vision. Surfaces and their intersections, curves and knots, three-dimensional manifolds, surfaces in dimension 4 etc., all these material are presented in an informal easy way, making the exposition available to undergraduate students. As to the pictures, they are really delightful. I especially enjoyed the movies of surfaces and movie moves. On the whole the book is a successful attempt of an introduction to topology focusing on its spirit and skipping its technical side.” Vladimir Turaev Directeur de Recherche au CNRS “This book is a definite enrichment to the literature in low-dimensional topology.” Mathematics Abstracts


Book Synopsis How Surfaces Intersect in Space by : J Scott Carter

Download or read book How Surfaces Intersect in Space written by J Scott Carter and published by World Scientific. This book was released on 1995-05-11 with total page 338 pages. Available in PDF, EPUB and Kindle. Book excerpt: This marvelous book of pictures illustrates the fundamental concepts of geometric topology in a way that is very friendly to the reader. The first chapter discusses the meaning of surface and space and gives the classification of orientable surfaces. In the second chapter we are introduced to the Möbius band and surfaces that can be constructed from this non-orientable piece of fabric. In chapter 3, we see how curves can fit in surfaces and how surfaces can fit into spaces with these curves on their boundary. Basic applications to knot theory are discussed and four-dimensional space is introduced. In Chapter 4 we learn about some 3-dimensional spaces and surfaces that sit inside them. These surfaces help us imagine the structures of the larger space. Chapter 5 is completely new! It contains recent results of Cromwell, Izumiya and Marar. One of these results is a formula relating the rank of a surface to the number of triple points. The other major result is a collection of examples of surfaces in 3-space that have one triple point and 6 branch points. These are beautiful generalizations of the Steiner Roman surface. Chapter 6 reviews the movie technique for examining surfaces in 4-dimensional space. Various movies of the Klein bottle are presented, and the Carter-Saito movie move theorem is explained. The author shows us how to turn the 2-sphere inside out by means of these movie moves and this illustration alone is well worth the price of the book! In the last chapter higher dimensional spaces are examined from an elementary point of view. This is a guide book to a wide variety of topics. It will be of value to anyone who wants to understand the subject by way of examples. Undergraduates, beginning graduate students, and non-professionals will profit from reading the book and from just looking at the pictures. Contents:Front MatterSurface and SpaceNon-orientable SurfacesCurves and KnotsOther Three Dimensional SpacesRelationshipsSurfaces in 4-DimensionsHigher Dimensional SpacesBack Matter Readership: Undergraduates, graduates and mathematicians. keywords:Moving Surfaces;Surfaces;Triple Point;Branch Points “In this excellent book the author teaches us to see a bit more than it meets our eyes. Without hurry he introduces us to the world of topological images. Step by step the reader learns the beauty of topological vision. Surfaces and their intersections, curves and knots, three-dimensional manifolds, surfaces in dimension 4 etc., all these material are presented in an informal easy way, making the exposition available to undergraduate students. As to the pictures, they are really delightful. I especially enjoyed the movies of surfaces and movie moves. On the whole the book is a successful attempt of an introduction to topology focusing on its spirit and skipping its technical side.” Vladimir Turaev Directeur de Recherche au CNRS “This book is a definite enrichment to the literature in low-dimensional topology.” Mathematics Abstracts

How Surfaces Intersect in Space

How Surfaces Intersect in Space

Author: J. Scott Carter

Publisher:

Published: 1995

Total Pages: 318

ISBN-13: 9780000098108

DOWNLOAD EBOOK

This marvelous book of pictures illustrates the fundamental concepts of geometric topology in a way that is very friendly to the reader. The first chapter discusses the meaning of surface and space and gives the classification of orientable surfaces. In the second chapter we are introduced to the Möbius band and surfaces that can be constructed from this non-orientable piece of fabric. In chapter 3, we see how curves can fit in surfaces and how surfaces can fit into spaces with these curves on their boundary. Basic applications to knot theory are discussed and four-dimensional space is introduced. In Chapter 4 we learn about some 3-dimensional spaces and surfaces that sit inside them. These surfaces help us imagine the structures of the larger space. Chapter 5 is completely new! It contains recent results of Cromwell, Izumiya and Marar. One of these results is a formula relating the rank of a surface to the number of triple points. The other major result is a collection of examples of surfaces in 3-space that have one triple point and 6 branch points. These are beautiful generalizations of the Steiner Roman surface. Chapter 6 reviews the movie technique for examining surfaces in 4-dimensional space. Various movies of the Klein bottle are presented, and the Carter-Saito movie move theorem is explained. The author shows us how to turn the 2-sphere inside out by means of these movie moves and this illustration alone is well worth the price of the book!In the last chapter higher dimensional spaces are examined from an elementary point of view. This is a guide book to a wide variety of topics. It will be of value to anyone who wants to understand the subject by way of examples. Undergraduates, beginning graduate students, and non-professionals will profit from reading the book and from just looking at the pictures.


Book Synopsis How Surfaces Intersect in Space by : J. Scott Carter

Download or read book How Surfaces Intersect in Space written by J. Scott Carter and published by . This book was released on 1995 with total page 318 pages. Available in PDF, EPUB and Kindle. Book excerpt: This marvelous book of pictures illustrates the fundamental concepts of geometric topology in a way that is very friendly to the reader. The first chapter discusses the meaning of surface and space and gives the classification of orientable surfaces. In the second chapter we are introduced to the Möbius band and surfaces that can be constructed from this non-orientable piece of fabric. In chapter 3, we see how curves can fit in surfaces and how surfaces can fit into spaces with these curves on their boundary. Basic applications to knot theory are discussed and four-dimensional space is introduced. In Chapter 4 we learn about some 3-dimensional spaces and surfaces that sit inside them. These surfaces help us imagine the structures of the larger space. Chapter 5 is completely new! It contains recent results of Cromwell, Izumiya and Marar. One of these results is a formula relating the rank of a surface to the number of triple points. The other major result is a collection of examples of surfaces in 3-space that have one triple point and 6 branch points. These are beautiful generalizations of the Steiner Roman surface. Chapter 6 reviews the movie technique for examining surfaces in 4-dimensional space. Various movies of the Klein bottle are presented, and the Carter-Saito movie move theorem is explained. The author shows us how to turn the 2-sphere inside out by means of these movie moves and this illustration alone is well worth the price of the book!In the last chapter higher dimensional spaces are examined from an elementary point of view. This is a guide book to a wide variety of topics. It will be of value to anyone who wants to understand the subject by way of examples. Undergraduates, beginning graduate students, and non-professionals will profit from reading the book and from just looking at the pictures.

Knots and Surfaces

Knots and Surfaces

Author: N. D. Gilbert

Publisher: Oxford University Press, UK

Published: 1994-12-01

Total Pages: 285

ISBN-13: 0191591904

DOWNLOAD EBOOK

Completely up-to-date, illustrated throughout, and written in an accessible style, Knots and Surfaces is an account of the mathematical theory of knots and its interaction with related fields. This is an area of intense research activity, and this text provides the advanced undergraduate with a superb introduction to this exciting field. Beginning with a simple diagrammatic approach, the book proceeds through recent advances to areas of current research. Topics including topological spaces, surfaces, the fundamental group, graphs, free groups, and group presentations combine to form a coherent and highly developed theory with which to explore and explain the accessible and intuitive problems of knots and surfaces. - ;The main theme of this book is the mathematical theory of knots and its interaction with the theory of surfaces and of group presentations. Beginning with a simple diagrammatic approach to the study of knots, reflecting the artistic and geometric appeal of interlaced forms, Knots and Surfaces takes the reader through recent advances in our understanding to areas of current research. Topics included are straightforward introductions to topological spaces, surfaces, the fundamental group, graphs, free groups, and group presentations. These topics combine into a coherent and highly developed theory to explore and explain the accessible and intuitive problems of knots and surfaces. Both as an introduction to several areas of prime importance to the development of pure mathematics today, and as an account of pure mathematics in action in an unusual context, this book presents novel challenges to students and other interested readers. -


Book Synopsis Knots and Surfaces by : N. D. Gilbert

Download or read book Knots and Surfaces written by N. D. Gilbert and published by Oxford University Press, UK. This book was released on 1994-12-01 with total page 285 pages. Available in PDF, EPUB and Kindle. Book excerpt: Completely up-to-date, illustrated throughout, and written in an accessible style, Knots and Surfaces is an account of the mathematical theory of knots and its interaction with related fields. This is an area of intense research activity, and this text provides the advanced undergraduate with a superb introduction to this exciting field. Beginning with a simple diagrammatic approach, the book proceeds through recent advances to areas of current research. Topics including topological spaces, surfaces, the fundamental group, graphs, free groups, and group presentations combine to form a coherent and highly developed theory with which to explore and explain the accessible and intuitive problems of knots and surfaces. - ;The main theme of this book is the mathematical theory of knots and its interaction with the theory of surfaces and of group presentations. Beginning with a simple diagrammatic approach to the study of knots, reflecting the artistic and geometric appeal of interlaced forms, Knots and Surfaces takes the reader through recent advances in our understanding to areas of current research. Topics included are straightforward introductions to topological spaces, surfaces, the fundamental group, graphs, free groups, and group presentations. These topics combine into a coherent and highly developed theory to explore and explain the accessible and intuitive problems of knots and surfaces. Both as an introduction to several areas of prime importance to the development of pure mathematics today, and as an account of pure mathematics in action in an unusual context, this book presents novel challenges to students and other interested readers. -

Knot Theory and Its Applications

Knot Theory and Its Applications

Author: Krishnendu Gongopadhyay

Publisher: American Mathematical Soc.

Published: 2016-09-21

Total Pages: 357

ISBN-13: 1470422573

DOWNLOAD EBOOK

This volume contains the proceedings of the ICTS program Knot Theory and Its Applications (KTH-2013), held from December 10–20, 2013, at IISER Mohali, India. The meeting focused on the broad area of knot theory and its interaction with other disciplines of theoretical science. The program was divided into two parts. The first part was a week-long advanced school which consisted of minicourses. The second part was a discussion meeting that was meant to connect the school to the modern research areas. This volume consists of lecture notes on the topics of the advanced school, as well as surveys and research papers on current topics that connect the lecture notes with cutting-edge research in the broad area of knot theory.


Book Synopsis Knot Theory and Its Applications by : Krishnendu Gongopadhyay

Download or read book Knot Theory and Its Applications written by Krishnendu Gongopadhyay and published by American Mathematical Soc.. This book was released on 2016-09-21 with total page 357 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the proceedings of the ICTS program Knot Theory and Its Applications (KTH-2013), held from December 10–20, 2013, at IISER Mohali, India. The meeting focused on the broad area of knot theory and its interaction with other disciplines of theoretical science. The program was divided into two parts. The first part was a week-long advanced school which consisted of minicourses. The second part was a discussion meeting that was meant to connect the school to the modern research areas. This volume consists of lecture notes on the topics of the advanced school, as well as surveys and research papers on current topics that connect the lecture notes with cutting-edge research in the broad area of knot theory.

Knot Theory and Its Applications

Knot Theory and Its Applications

Author: Kunio Murasugi

Publisher: Springer Science & Business Media

Published: 2009-12-29

Total Pages: 348

ISBN-13: 0817647198

DOWNLOAD EBOOK

This book introduces the study of knots, providing insights into recent applications in DNA research and graph theory. It sets forth fundamental facts such as knot diagrams, braid representations, Seifert surfaces, tangles, and Alexander polynomials. It also covers more recent developments and special topics, such as chord diagrams and covering spaces. The author avoids advanced mathematical terminology and intricate techniques in algebraic topology and group theory. Numerous diagrams and exercises help readers understand and apply the theory. Each chapter includes a supplement with interesting historical and mathematical comments.


Book Synopsis Knot Theory and Its Applications by : Kunio Murasugi

Download or read book Knot Theory and Its Applications written by Kunio Murasugi and published by Springer Science & Business Media. This book was released on 2009-12-29 with total page 348 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book introduces the study of knots, providing insights into recent applications in DNA research and graph theory. It sets forth fundamental facts such as knot diagrams, braid representations, Seifert surfaces, tangles, and Alexander polynomials. It also covers more recent developments and special topics, such as chord diagrams and covering spaces. The author avoids advanced mathematical terminology and intricate techniques in algebraic topology and group theory. Numerous diagrams and exercises help readers understand and apply the theory. Each chapter includes a supplement with interesting historical and mathematical comments.

New Ideas In Low Dimensional Topology

New Ideas In Low Dimensional Topology

Author: Vassily Olegovich Manturov

Publisher: World Scientific

Published: 2015-01-27

Total Pages: 540

ISBN-13: 9814630632

DOWNLOAD EBOOK

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.


Book Synopsis New Ideas In Low Dimensional Topology by : Vassily Olegovich Manturov

Download or read book New Ideas In Low Dimensional Topology written by Vassily Olegovich Manturov and published by World Scientific. This book was released on 2015-01-27 with total page 540 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.