Braid group

The braid group is the group whose elements are n -strand braids. The group operation is the concatenation of braids and the neutral element is the n- braid with no crossings.

There is for every natural number n is a braid group. Braid groups are examined in the mathematical field of the topology. Braid groups were first defined in the article theory of braids from 1925 by Emil Artin; a similar construction but there were also early as 1891 in a work by Adolf Hurwitz.

Geometric definition

A -stranded braid is a set of non-intersecting curves ( with ) in that start in, in forming and in their parameterization, the third coordinate function ( in the figure, the z- coordinate) is monotonically increasing. These curves are called strands.

Each n- braid it situates an element of the symmetric group: the permutation is defined by the fact that one follows the i-th strand to its end point. The core of this picture is the so-called pure braid group. Thus, it consists only of such pigtails, in which the i-th strand ends at position i.

And two braids are equivalent if they are isotope, that is, when there is a continuous family of pigtails, which starts and ends in.

Group Properties

The set of all ( equivalence classes of ) - strand braids created a group. The linkage is the attachment of a braid under the other, wherein the coordinate is rescaled. The identity element of the group is the braid with parallel strands. The inverse element of a braid is just its mirror.

One can represent each plait as a result of over- or under- crossings of the strands. These are just the generator shown in the figure or.

One can get in a sketch to illustrate that each producer multiplied by its inverse yields the identity element.

Presentation by generators and relations

The braid group has the following presentation by generators and relations:

Producers:

• .

Relations:

• For
• For

This definition is algebraic with the geometric equivalent.

In particular, braid groups are a special case of Artin groups.

Examples

The braid group consists of only one element. The braid group is the infinite cyclic group. The braid group has the representation

And is non- commutative.

Braid groups as mapping class groups

The mapping class group of the circular disk with marked points is isomorphic to the braid group.

Pure braid group

Each n -stranded braid determines a permutation of n elements. The pure braid group is the core of the so- defined homomorphism.

Applications

Mathematician primarily interested in the application of knot theory: By connecting the upper end of the braid with the lower end, you get a tangle. Equivalents braids generate equivalent tangles. On the other hand, any entanglement be brought about by isotopic deformation in the form of a closed braid ( set of Alexander). When create two pigtails same entanglement, clears the set of Markov ( Andrei Markov, 1903-1979, son of Andrei Markov, 1856-1922 ).