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In this well-written presentation, motivated by numerous examples and problems, the authors introduce the basic theory of braid groups, highlighting several definitions that show their equivalence; this is followed by a treatment of the relationship between braids, knots and links. Important results then treat the linearity and orderability of the subject. Relevant additional material is included in five large appendices. Braid Groups will serve graduate students and a number of mathematicians coming from diverse disciplines.
Book Synopsis Braid Groups by : Christian Kassel
Download or read book Braid Groups written by Christian Kassel and published by Springer Science & Business Media. This book was released on 2008-06-28 with total page 349 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this well-written presentation, motivated by numerous examples and problems, the authors introduce the basic theory of braid groups, highlighting several definitions that show their equivalence; this is followed by a treatment of the relationship between braids, knots and links. Important results then treat the linearity and orderability of the subject. Relevant additional material is included in five large appendices. Braid Groups will serve graduate students and a number of mathematicians coming from diverse disciplines.
This book covers basic properties of complex reflection groups, such as characterization, Steinberg theorem, Gutkin-Opdam matrices, Solomon theorem and applications, including the basic findings of Springer theory on eigenspaces.
Book Synopsis Introduction to Complex Reflection Groups and Their Braid Groups by : Michel Broué
Download or read book Introduction to Complex Reflection Groups and Their Braid Groups written by Michel Broué and published by Springer. This book was released on 2010-01-28 with total page 150 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book covers basic properties of complex reflection groups, such as characterization, Steinberg theorem, Gutkin-Opdam matrices, Solomon theorem and applications, including the basic findings of Springer theory on eigenspaces.
This is the award-winning monograph of the Sunyer i Balaguer Prize 1999. The book presents recently discovered connections between Artin’s braid groups and left self-distributive systems, which are sets equipped with a binary operation satisfying the identity x(yz) = (xy)(xz). Although not a comprehensive course, the exposition is self-contained, and many basic results are established. In particular, the first chapters include a thorough algebraic study of Artin’s braid groups.
Book Synopsis Braids and Self-Distributivity by : Patrick Dehornoy
Download or read book Braids and Self-Distributivity written by Patrick Dehornoy and published by Birkhäuser. This book was released on 2012-12-06 with total page 637 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the award-winning monograph of the Sunyer i Balaguer Prize 1999. The book presents recently discovered connections between Artin’s braid groups and left self-distributive systems, which are sets equipped with a binary operation satisfying the identity x(yz) = (xy)(xz). Although not a comprehensive course, the exposition is self-contained, and many basic results are established. In particular, the first chapters include a thorough algebraic study of Artin’s braid groups.
In Chapter 6, we describe the concept of braid equivalence from the topological point of view. This will lead us to a new concept braid homotopy that is discussed fully in the next chapter. As just mentioned, in Chapter 7, we shall discuss the difference between braid equivalence and braid homotopy. Also in this chapter, we define a homotopy braid invariant that turns out to be the so-called Milnor number. Chapter 8 is a quick review of knot theory, including Alexander's theorem. While, Chapters 9 is devoted to Markov's theorem, which allows the application of this theory to other fields. This was one of the motivations Artin had in mind when he began studying braid theory. In Chapter 10, we discuss the primary applications of braid theory to knot theory, including the introduction of the most important invariants of knot theory, the Alexander polynomial and the Jones polynomial. In Chapter 11, motivated by Dirac's string problem, the ordinary braid group is generalized to the braid groups of various surfaces. We discuss these groups from an intuitive and diagrammatic point of view. In the last short chapter 12, we present without proof one theorem, due to Gorin and Lin [GoL] , that is a surprising application of braid theory to the theory of algebraic equations.
Book Synopsis A Study of Braids by : Kunio Murasugi
Download or read book A Study of Braids written by Kunio Murasugi and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 287 pages. Available in PDF, EPUB and Kindle. Book excerpt: In Chapter 6, we describe the concept of braid equivalence from the topological point of view. This will lead us to a new concept braid homotopy that is discussed fully in the next chapter. As just mentioned, in Chapter 7, we shall discuss the difference between braid equivalence and braid homotopy. Also in this chapter, we define a homotopy braid invariant that turns out to be the so-called Milnor number. Chapter 8 is a quick review of knot theory, including Alexander's theorem. While, Chapters 9 is devoted to Markov's theorem, which allows the application of this theory to other fields. This was one of the motivations Artin had in mind when he began studying braid theory. In Chapter 10, we discuss the primary applications of braid theory to knot theory, including the introduction of the most important invariants of knot theory, the Alexander polynomial and the Jones polynomial. In Chapter 11, motivated by Dirac's string problem, the ordinary braid group is generalized to the braid groups of various surfaces. We discuss these groups from an intuitive and diagrammatic point of view. In the last short chapter 12, we present without proof one theorem, due to Gorin and Lin [GoL] , that is a surprising application of braid theory to the theory of algebraic equations.
In the present treatise progress in topological approach to Hall system physics is reported, including recent achievements in graphene. The new homotopy methods of cyclotron braid subgroups, originally introduced by the authors, turn out to be of particular convenience in order to grasp peculiarity of 2D charged systems upon magnetic field resulting in fractional Hall states. The identified cyclotron braids allow for natural recovery of Laughlin correlations from the first principles, without invoking any artificial constructions as composite fermions with flux tubes or vortices. Progress in understanding of the structure and role of composite fermions in Hall system is provided, which can also lead to some corrections of numerical results in energy minimization made within the traditional formulation of the composite fermion model. The crucial significance of carrier mobility, apart from interaction in creation of the fractional quantum Hall effect (FQHE), is described and supported by recent graphene experiments. Recent advancement in the FQHE field including topological insulators and optical lattices is reviewed and commented upon in terms of the braid group approach. The braid group methods are presented from a more general point of view including proposition of pure braid group application. Book jacket.
Book Synopsis Application of Braid Groups in 2D Hall System Physics by : Janusz Jacak
Download or read book Application of Braid Groups in 2D Hall System Physics written by Janusz Jacak and published by World Scientific. This book was released on 2012 with total page 160 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the present treatise progress in topological approach to Hall system physics is reported, including recent achievements in graphene. The new homotopy methods of cyclotron braid subgroups, originally introduced by the authors, turn out to be of particular convenience in order to grasp peculiarity of 2D charged systems upon magnetic field resulting in fractional Hall states. The identified cyclotron braids allow for natural recovery of Laughlin correlations from the first principles, without invoking any artificial constructions as composite fermions with flux tubes or vortices. Progress in understanding of the structure and role of composite fermions in Hall system is provided, which can also lead to some corrections of numerical results in energy minimization made within the traditional formulation of the composite fermion model. The crucial significance of carrier mobility, apart from interaction in creation of the fractional quantum Hall effect (FQHE), is described and supported by recent graphene experiments. Recent advancement in the FQHE field including topological insulators and optical lattices is reviewed and commented upon in terms of the braid group approach. The braid group methods are presented from a more general point of view including proposition of pure braid group application. Book jacket.
This manuscript is devoted to classifying the isomorphism classes of the virtually cyclic subgroups of the braid groups of the 2-sphere. As well as enabling us to understand better the global structure of these groups, it marks an important step in the computation of the K-theory of their group rings. The classification itself is somewhat intricate, due to the rich structure of the finite subgroups of these braid groups, and is achieved by an in-depth analysis of their group-theoretical and topological properties, such as their centralisers, normalisers and cohomological periodicity. Another important aspect of our work is the close relationship of the braid groups with mapping class groups. This manuscript will serve as a reference for the study of braid groups of low-genus surfaces, and isaddressed to graduate students and researchers in low-dimensional, geometric and algebraic topology and in algebra.
Book Synopsis The Classification of the Virtually Cyclic Subgroups of the Sphere Braid Groups by : Daciberg Lima Goncalves
Download or read book The Classification of the Virtually Cyclic Subgroups of the Sphere Braid Groups written by Daciberg Lima Goncalves and published by Springer Science & Business Media. This book was released on 2013-09-08 with total page 102 pages. Available in PDF, EPUB and Kindle. Book excerpt: This manuscript is devoted to classifying the isomorphism classes of the virtually cyclic subgroups of the braid groups of the 2-sphere. As well as enabling us to understand better the global structure of these groups, it marks an important step in the computation of the K-theory of their group rings. The classification itself is somewhat intricate, due to the rich structure of the finite subgroups of these braid groups, and is achieved by an in-depth analysis of their group-theoretical and topological properties, such as their centralisers, normalisers and cohomological periodicity. Another important aspect of our work is the close relationship of the braid groups with mapping class groups. This manuscript will serve as a reference for the study of braid groups of low-genus surfaces, and isaddressed to graduate students and researchers in low-dimensional, geometric and algebraic topology and in algebra.
The central theme of this study is Artin's braid group and the many ways that the notion of a braid has proved to be important in low-dimensional topology. In Chapter 1 the author is concerned with the concept of a braid as a group of motions of points in a manifold. She studies structural and algebraic properties of the braid groups of two manifolds, and derives systems of defining relations for the braid groups of the plane and sphere. In Chapter 2 she focuses on the connections between the classical braid group and the classical knot problem. After reviewing basic results she proceeds to an exploration of some possible implications of the Garside and Markov theorems. Chapter 3 offers discussion of matrix representations of the free group and of subgroups of the automorphism group of the free group. These ideas come to a focus in the difficult open question of whether Burau's matrix representation of the braid group is faithful. Chapter 4 is a broad view of recent results on the connections between braid groups and mapping class groups of surfaces. Chapter 5 contains a brief discussion of the theory of "plats." Research problems are included in an appendix.
Book Synopsis Braids, Links, and Mapping Class Groups by : Joan S. Birman
Download or read book Braids, Links, and Mapping Class Groups written by Joan S. Birman and published by Princeton University Press. This book was released on 1974 with total page 244 pages. Available in PDF, EPUB and Kindle. Book excerpt: The central theme of this study is Artin's braid group and the many ways that the notion of a braid has proved to be important in low-dimensional topology. In Chapter 1 the author is concerned with the concept of a braid as a group of motions of points in a manifold. She studies structural and algebraic properties of the braid groups of two manifolds, and derives systems of defining relations for the braid groups of the plane and sphere. In Chapter 2 she focuses on the connections between the classical braid group and the classical knot problem. After reviewing basic results she proceeds to an exploration of some possible implications of the Garside and Markov theorems. Chapter 3 offers discussion of matrix representations of the free group and of subgroups of the automorphism group of the free group. These ideas come to a focus in the difficult open question of whether Burau's matrix representation of the braid group is faithful. Chapter 4 is a broad view of recent results on the connections between braid groups and mapping class groups of surfaces. Chapter 5 contains a brief discussion of the theory of "plats." Research problems are included in an appendix.
The central theme of this study is Artin's braid group and the many ways that the notion of a braid has proved to be important in low-dimensional topology. In Chapter 1 the author is concerned with the concept of a braid as a group of motions of points in a manifold. She studies structural and algebraic properties of the braid groups of two manifolds, and derives systems of defining relations for the braid groups of the plane and sphere. In Chapter 2 she focuses on the connections between the classical braid group and the classical knot problem. After reviewing basic results she proceeds to an exploration of some possible implications of the Garside and Markov theorems. Chapter 3 offers discussion of matrix representations of the free group and of subgroups of the automorphism group of the free group. These ideas come to a focus in the difficult open question of whether Burau's matrix representation of the braid group is faithful. Chapter 4 is a broad view of recent results on the connections between braid groups and mapping class groups of surfaces. Chapter 5 contains a brief discussion of the theory of "plats." Research problems are included in an appendix.
Book Synopsis Braids, Links, and Mapping Class Groups. (AM-82), Volume 82 by : Joan S. Birman
Download or read book Braids, Links, and Mapping Class Groups. (AM-82), Volume 82 written by Joan S. Birman and published by Princeton University Press. This book was released on 2016-03-02 with total page 237 pages. Available in PDF, EPUB and Kindle. Book excerpt: The central theme of this study is Artin's braid group and the many ways that the notion of a braid has proved to be important in low-dimensional topology. In Chapter 1 the author is concerned with the concept of a braid as a group of motions of points in a manifold. She studies structural and algebraic properties of the braid groups of two manifolds, and derives systems of defining relations for the braid groups of the plane and sphere. In Chapter 2 she focuses on the connections between the classical braid group and the classical knot problem. After reviewing basic results she proceeds to an exploration of some possible implications of the Garside and Markov theorems. Chapter 3 offers discussion of matrix representations of the free group and of subgroups of the automorphism group of the free group. These ideas come to a focus in the difficult open question of whether Burau's matrix representation of the braid group is faithful. Chapter 4 is a broad view of recent results on the connections between braid groups and mapping class groups of surfaces. Chapter 5 contains a brief discussion of the theory of "plats." Research problems are included in an appendix.
Since the discovery that Artin's braid groups enjoy a left-invariant linear ordering, several different approaches have been used to understand this phenomenon. This text provides an account of those approaches, involving varied objects & domains as combinatorial group theory, self-distributive algebra & finite combinatorics.
Book Synopsis Ordering Braids by : Patrick Dehornoy
Download or read book Ordering Braids written by Patrick Dehornoy and published by American Mathematical Soc.. This book was released on 2008 with total page 339 pages. Available in PDF, EPUB and Kindle. Book excerpt: Since the discovery that Artin's braid groups enjoy a left-invariant linear ordering, several different approaches have been used to understand this phenomenon. This text provides an account of those approaches, involving varied objects & domains as combinatorial group theory, self-distributive algebra & finite combinatorics.
Contents:Notes on Subfactors and Statistical Mechanics (V F R Jones)Polynomial Invariants in Knot Theory (L H Kauffman)Algebras of Loops on Surfaces, Algebras of Knots, and Quantization (V G Turaev)Quantum Groups (L Faddeev et al.)Introduction to the Yang-Baxter Equation (M Jimbo)Integrable Systems Related to Braid Groups and Yang-Baxter Equation (T Kohno)The Yang-Baxter Relation: A New Tool for Knot Theory (Y Akutsu et al.)Akutsu-Wadati Link Polynomials from Feynman-Kauffman Diagrams (M-L Ge et al.)Quantum Field Theory and the Jones Polynomial (E Witten) Readership: Mathematical physicists.
Book Synopsis Braid Group, Knot Theory and Statistical Mechanics by : C N Yang
Download or read book Braid Group, Knot Theory and Statistical Mechanics written by C N Yang and published by World Scientific. This book was released on 1991-06-05 with total page 336 pages. Available in PDF, EPUB and Kindle. Book excerpt: Contents:Notes on Subfactors and Statistical Mechanics (V F R Jones)Polynomial Invariants in Knot Theory (L H Kauffman)Algebras of Loops on Surfaces, Algebras of Knots, and Quantization (V G Turaev)Quantum Groups (L Faddeev et al.)Introduction to the Yang-Baxter Equation (M Jimbo)Integrable Systems Related to Braid Groups and Yang-Baxter Equation (T Kohno)The Yang-Baxter Relation: A New Tool for Knot Theory (Y Akutsu et al.)Akutsu-Wadati Link Polynomials from Feynman-Kauffman Diagrams (M-L Ge et al.)Quantum Field Theory and the Jones Polynomial (E Witten) Readership: Mathematical physicists.
