scholarly journals Notes on the Topology of Hyperplane Arrangements and Braid Groups

2018 ◽  
Author(s):  
Claudio Procesi
Keyword(s):  
Braid Groups ◽  
Hyperplane Arrangements

HOPF-BRAID GROUPS

1998 ◽  
Vol 07 (08) ◽  
pp. 1107-1117
Author(s):  
VINCENT MOULTON
Keyword(s):  
Braid Group ◽  
Braid Groups ◽  
Hyperplane Arrangements ◽  
Motion Group ◽  
Three Sphere ◽  
Group A ◽  
Complex Lines ◽  
Group Of Motions ◽  
Hopf Link

In this note we define the Hopf-braid group, a group that is directly related to the group of motions of n mutually distinct lines through the origin in [Formula: see text], which is better known as the braid group of the two-sphere. It is also related to the motion group of the Hopf link in the three-sphere. This relationship is provided by considering the link of a union of complex lines through the origin in [Formula: see text] (i.e. the intersection of the lines with the unit 3-sphere centered at the origin in [Formula: see text]). Through the study of this group we also illustrate some of the connections between the field of knots and braids and that of hyperplane arrangements.

Braids

2017 ◽  
Author(s):  
Aaron Abrams
Keyword(s):  
Group Theory ◽  
Knot Theory ◽  
Braid Groups ◽  
Hyperplane Arrangements ◽  
Configuration Spaces ◽  
Research Projects ◽  
And Topology ◽  
Mathematics And Science

This chapter deals with mathematical braids, an idealized abstraction of the familiar hair and bread braids. In a mathematical braid, the “strings” remain separate at the ends rather than being fused together. Furthermore, a mathematical braid can have any number of braided strings and the braiding can occur in any pattern. This chapter gives different ways to think about mathematical braids and some of the basic theorems about them. It also describes several ways that braids relate to other parts of mathematics and science such as robotics, knot theory, and hyperplane arrangements. After providing an overview of some group theory relating to braids, the chapter considers configuration spaces that connect braid groups and topology as well as the concept of punctured disks. Finally, it presents an experiment for braiding the hair by first making a ponytail and then doing the braiding. The discussion includes exercises and research projects.

scholarly journals An Ordering for Groups of Pure Braids and Fibre-Type Hyperplane Arrangements

2003 ◽  
Vol 55 (4) ◽  
pp. 822-838 ◽  
Author(s):  
Djun Maximilian Kim ◽  
Dale Rolfsen
Keyword(s):  
Fibre Type ◽  
Free Groups ◽  
Braid Groups ◽  
Hyperplane Arrangements ◽  
The Other ◽  
Fundamental Groups ◽  
Magnus Expansion ◽  
Direct Generalization ◽  
Complex Hyperplane

AbstractWe define a total ordering of the pure braid groups which is invariant under multiplication on both sides. This ordering is natural in several respects. Moreover, it well-orders the pure braids which are positive in the sense of Garside. The ordering is defined using a combination of Artin's combing technique and the Magnus expansion of free groups, and is explicit and algorithmic.By contrast, the full braid groups (on 3 or more strings) can be ordered in such a way as to be invariant on one side or the other, but not both simultaneously. Finally, we remark that the same type of ordering can be applied to the fundamental groups of certain complex hyperplane arrangements, a direct generalization of the pure braid groups.

Pseudo-Anosov Theory

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Braid Groups ◽  
Intersection Numbers ◽  
Branched Covers ◽  
Algebraic Integers ◽  
Dehn Twists ◽  
Mapping Classes ◽  
Dynamical Properties

This chapter focuses on the construction as well as the algebraic and dynamical properties of pseudo-Anosov homeomorphisms. It first presents five different constructions of pseudo-Anosov mapping classes: branched covers, constructions via Dehn twists, homological criterion, Kra's construction, and a construction for braid groups. It then proves a few fundamental facts concerning stretch factors of pseudo-Anosov homeomorphisms, focusing on the theorem that pseudo-Anosov stretch factors are algebraic integers. It also considers the spectrum of pseudo-Anosov stretch factors, along with the special properties of those measured foliations that are the stable (or unstable) foliations of some pseudo-Anosov homeomorphism. Finally, it describes the orbits of a pseudo-Anosov homeomorphism as well as lengths of curves and intersection numbers under iteration.

New Signature Scheme over the Braid Groups

2011 ◽  
Vol 32 (12) ◽  
pp. 2930-2934
Author(s):  
Yun Wei ◽  
Guo-hua Xiong ◽  
Wan-su Bao ◽  
Xing-kai Zhang
Keyword(s):  
Braid Groups ◽  
Signature Scheme

scholarly journals Crystallographic groups and flat manifolds from surface braid groups

2020 ◽  
pp. 107560
Author(s):  
Daciberg Lima Gonçalves ◽  
John Guaschi ◽  
Oscar Ocampo ◽  
Carolina de Miranda e Pereiro
Keyword(s):  
Braid Groups ◽  
Crystallographic Groups ◽  
Flat Manifolds

scholarly journals The $$R_\infty $$ property for pure Artin braid groups

2021 ◽  
Vol 195 (1) ◽  
pp. 15-33
Author(s):  
Karel Dekimpe ◽  
Daciberg Lima Gonçalves ◽  
Oscar Ocampo
Keyword(s):  
Braid Groups

scholarly journals On polytopes and generalizations of the KLT relations

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Nikhil Kalyanapuram
Keyword(s):  
Scattering Amplitudes ◽  
Large Class ◽  
Moduli Space ◽  
Differential Forms ◽  
Natural Generalization ◽  
Intersection Theory ◽  
Hyperplane Arrangements ◽  
Number Of Solutions ◽  
Scalar Theories ◽  
Double Copy

Abstract We combine the technology of the theory of polytopes and twisted intersection theory to derive a large class of double copy relations that generalize the classical relations due to Kawai, Lewellen and Tye (KLT). To do this, we first study a generalization of the scattering equations of Cachazo, He and Yuan. While the scattering equations were defined on ℳ0, n — the moduli space of marked Riemann spheres — the new scattering equations are defined on polytopes known as accordiohedra, realized as hyperplane arrangements. These polytopes encode as patterns of intersection the scattering amplitudes of generic scalar theories. The twisted period relations of such intersection numbers provide a vast generalization of the KLT relations. Differential forms dual to the bounded chambers of the hyperplane arrangements furnish a natural generalization of the Bern-Carrasco-Johansson (BCJ) basis, the number of which can be determined by counting the number of solutions of the generalized scattering equations. In this work the focus is on a generalization of the BCJ expansion to generic scalar theories, although we use the labels KLT and BCJ interchangeably.

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