The braid group: redefining (original) (raw)
The concept of free group based on braid group
Journal of Physics: Conference Series, 2018
Although free groups have long been present as part of algebra, and it steady has its implications for each implementation target, especially to determine invariant. However, the decrease in free groups of webbing requires different meanings of braid. This paper reveals a concept to form a free group of braid groups.
We introduce braids via their historical roots and uses, make connections with knot theory and present the mathematical theory of braids through the braid group. Several basic mathematical properties of braids are explored and equivalence problems under several conditions defined and partly solved. The connection with knots is spelled out in detail and translation methods are presented. Finally a number of applications of braid theory are given. The presentation is pedagogical and principally aimed at interested readers from different fields of mathematics and natural science. The discussions are as self-contained as can be expected within the space limits and require very little previous mathematical knowledge. Literature references are given throughout to the original papers and to overview sources where more can be learned.
New Permutation Representations of the Braid Group
Arxiv preprint arXiv: …, 2008
We give a new infinite family of group homomorphisms from the braid group B k to the symmetric group S mk for all k and m ≥ 2. Most known permutation representations of braids are included in this family. We prove that the homomorphisms in this family are non-cyclic and transitive. For any divisor l of m, 1 ≤ l < m, we prove in particular that if m l is odd then there are 1+ m l non-conjugate homomorphisms included in our family. We define a certain natural restriction on homomorphisms B k → Sn, common to all homomorphisms in our family, which we term good, and of which there are two types. We prove that all good homomorphisms B k → S mk of type 1 are included in the infinite family of homomorphisms we gave. For m = 3, we prove that all good homomorphisms B k → S 3k of type 2 are also included in this family. Finally, we refute a conjecture made in [MaSu05] regarding permutation representations of braids and give an updated conjecture.
The pure braid groups and their relatives
In this talk, I discuss the resonance varieties, the lower central series ranks, the Chen ranks, and the formality properties of several families of braid-like groups: the pure braid groups P_n, the welded pure braid groups wP_n, the virtual pure braid groups vP_n, as well as their `upper' variants, wP_n^+ and vP_n^+. I will also discuss several natural homomorphisms between these groups, and various ways to distinguish among the pure braid groups and their relatives. This is joint work with He Wang.
A New Algorithm for Solving the Word Problem in Braid Groups
Advances in Mathematics, 2002
One of the most interesting questions about a group is if its word problem can be solved and how. The word problem in the braid group is of particular interest to topologists, algebraists and geometers, and is the target of intensive current research. We look at the braid group from a topological point of view (rather than a geometrical one). The braid group is defined by the action of diffeomorphisms on the fundamental group of a punctured disk. We exploit the topological definition of the braid group in order to give a new approach for solving its word problem. Our algorithm is faster, in comparison with known algorithms, for short braid words with respect to the number of generators combining the braid, and it is almost independent of the number of strings in the braids. Moreover, the algorithm is based on a new computer presentation of the elements of the fundamental group of a punctured disk. This presentation can be used also for other algorithms.
Fast Nielsen–Thurston classification of braids
Algebraic & Geometric Topology, 2014
We prove the existence of an algorithm which solves the reducibility problem in braid groups and runs in quadratic time with respect to the braid length for any fixed braid index.
Algorithmic Problems in the Braid Groups
University College London, Ph.D. Thesis, 2002
We introduce the braid groups in their connection to knot theory and investigate several of their properties. Based on term rewriting systems, which we review, we find new solutions to the word and conjugacy problems in the braid groups. A similar problem asks for the minimal length word for an equivalence class in a given braid group which we prove to be NP-complete (after a review of this concept) and present a new algorithm for it. As this algorithm takes an exponentially increasing amount of time, we construct an algebraic approximation algorithm which we find to work well. We consider several methods of approximating the minimal word via computer simulation of the braid strings moving under the influence of certain forces. Using the theory of tangles which we also review, we construct a new notation for knots which is usable by a computer. From this notation, we construct an efficient algorithm to find the braid or plat whose closure is ambient isotopic to any given knot. Finally, we apply the computer software developed for these problems to the solar coronal heating problem by simulating magnetic flux tubes. We also present a number of incidental results that were found along the way of researching these problems.