InfoIn mathematics, the braid group on strands, denoted by, is a group which has an intuitive geometrical representation, and in a sense generalizes the symmetric group, Here, is a natural number; if, then is an infinite group.more...Braid groups find applications in knot theory, since any knot may be represented as the closure of certain braids.

This fact is also related to the braid groups well known in knot theory.The relation can be understood when one considers the fact that in two dimensions the group of permutations of two particles is no longer the symmetric group S 2 (with two elements) but rather the braid group B 2 (with an infinite number of elements).…

One is a very tame group and is isomorphic to the mapping class group of the infinitely punctured disk -- a discrete set of punctures limiting to the boundary of the disk.The second group can be thought of the same as with finite braid groups.Place a strand at each of the points and the set of all braids -- where a braid is defined to be a collection of paths from the points to the points so that the function yields a permutation on endpoints -- is isomorphic to this wilder group.…

The word problem is also efficiently solved via the Lawrence-Krammer representation.Since there are nevertheless several hard computational problems about braid groups, applications in cryptography have been suggested.In analogy with the action of the symmetric group by permutations, in various mathematical settings there exists a natural action the braid group on -tuples of objects or on the -folded tensor product that involves some "twists".…

Her dissertation was titled Braid groups and their relationship to mapping class groups.

These are just the relations for the infinite braid group, together with the relations u = 0.Similarly one can define a nil-Coxeter algebra for any Coxeter system, by adding the relations u = 0 to the relations of the corresponding generalized braid group.…

This sequence splits and therefore pure braid groups are realized as iterated semi-direct products of free groups.

Such structures play an important role in modern mathematical physics and lead to quantum knot invariants.Elements of the braid group can be represented more concretely by matrices.One classical such representation is Burau representation, where the matrix entries are single variable Laurent polynomials.…

A Garside group is biautomatic and hence has soluble word problem and conjugacy problem.Examples of such groups include braid groups and, more generally, Artin groups of finite Coxeter type.The name was coined by Dehornoy and Paris to mark the work of F. A. Garside on the conjugacy problem for braid groups.…

In mathematics and theoretical physics, braid statistics is a generalization of the statistics of bosons and fermions based on the concept of braid group.A similar notion exists using a loop braid group.…

Such groups may be described by explicit presentations, as was shown by, For an elementary treatment along these lines, see the article on braid groups.Braid groups may also be given a deeper mathematical interpretation: as the fundamental group of certain configuration spaces.To explain how to reduce a braid group in the sense of Artin to a fundamental group, we consider a connected manifold X of dimension at least 2.…

Other, related, mathematical rings are called Hecke algebras, although the link to Hecke operators is not entirely obvious.These algebras include certain quotients of the group algebras of braid groups.The presence of this commutative operator algebra plays a significant role in the harmonic analysis of modular forms and generalisations.…

In particular Diffie-Hellman key exchange uses finite cyclic groups.So the term group-based cryptography refers mostly to cryptographic protocols that use infinite nonabelian groups such as a braid group.…

The idea is that braids can be organized into groups, in which the group operation is 'do the first braid on a set of strings, and then follow it with a second on the twisted strings'.Such groups may be described by explicit presentations, as was shown by, For an elementary treatment along these lines, see the article on braid groups.Braid groups may also be given a deeper mathematical interpretation: as the fundamental group of certain configuration spaces.…

Usually, the cables themselves are with a knit stitch while the background is done in purl.As the number of cables increases, the number of crossing patterns increases, as described by the braid group.Various visual effects are also possible by shifting the center lines of the undulating cables, or by changing the space between the cables, making them denser or more open.…

The term braided comes from the fact that the braid group plays an important role in the theory of braided monoidal categories.Partly for this reason, braided monoidal categories and various related notions are important in the theory of knot invariants.…

There is also a package called CHEVIE for GAP3 with special support for braid groups.The word problem is also efficiently solved via the Lawrence-Krammer representation.…

In papers published in 2000 and 2001, Daan Krammer and Stephen Bigelow established the faithfulness of Lawrence's representation.This result goes by the phrase "braid groups are linear."…

Dehornoy's original discovery of the order on the braid group used huge cardinals, but there are now several more elementary constructions of it.

The loop braid group is a mathematical group structure that is used in some models of theoretical physics to model the exchange of particles with loop-like topologies within three dimensions of space and time.The basic operations which generate a loop braid group for n loops are exchanges of two adjacent loops, and passing one adjacent loop through another.The topology forces these generators to satisfy some relations, which determine the group.…