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 without crossovers. ${\displaystyle B_{n}}$

There is a braid group for every natural number n . Braid groups are studied in the mathematical field of topology . Braid groups were first defined in the article Theory of Braids from 1925 by Emil Artin ; There was a similar construction in 1891 in a work by Adolf Hurwitz . ${\displaystyle B_{n}}$

A braid in three-dimensional space

Geometric definition

A -string braid is a set of non-intersecting curves (with ) in , which begin in, end in and in whose parameterization the third coordinate function (the z-coordinate in the figure) is monotonically increasing. These curves are called strands . ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle k_{i}}$ ${\displaystyle 1\leq i\leq n}$ ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle \left({\frac {i}{n}},0,0\right)}$ ${\displaystyle \left({\frac {\sigma (i)}{n}},0,1\right)}$

An element of the symmetrical group is assigned to each n-pigtail as follows : the permutation is defined by following the i-th strand to its end point . The core of this figure is the so-called pure braid group . It consists only of those braids in which the i-th strand ends at position i. ${\displaystyle \sigma }$ ${\displaystyle S_{n}}$ ${\displaystyle \sigma }$ ${\displaystyle \sigma (i)}$

Two braids and are equivalent when they are isotopic , that is, when there is a continuous family of braids that starts and ends in. ${\displaystyle b_{0}}$ ${\displaystyle b_{1}}$ ${\displaystyle b_{t},\,t\in (0,1)}$ ${\displaystyle b_{0}}$ ${\displaystyle b_{1}}$

The producers and ${\displaystyle \sigma _{i}^{-1}}$ ${\displaystyle \sigma _{i}^{}}$

Group properties

The set of all (equivalence classes of) -string braids creates a group . The link is to add one braid under the other, rescaling the coordinate. The single element of the group is the braid with parallel strands. The inverse element of a braid is precisely its reflection . ${\displaystyle n}$ ${\displaystyle z}$ ${\displaystyle n}$

Each braid can be represented as a sequence of crossings or crossings of the strands. These are the producers or shown in the figure . ${\displaystyle \sigma _{i}^{\,}}$ ${\displaystyle \sigma _{i}^{-1}}$

One can illustrate in a sketch that each generator multiplied by its inverse results in the neutral element. ${\displaystyle \sigma _{i}^{}}$ ${\displaystyle \sigma _{i}^{-1}}$

Representation by producers and relations

The braid group has the following representation by means of producers and relations : ${\displaystyle B_{n}}$

Producer:

• ${\displaystyle \sigma _{1}^{},\ldots ,\sigma _{n-1}^{}}$.

Relations:

• ${\displaystyle \sigma _{i}\sigma _{j}=\sigma _{j}\sigma _{i}\quad }$   For   ${\displaystyle |i-j|\geq 2}$
• ${\displaystyle \sigma _{i}\sigma _{i+1}\sigma _{i}=\sigma _{i+1}\sigma _{i}\sigma _{i+1}\quad }$   For   ${\displaystyle 1\leq i\leq n-2\,.}$

(The last relation goes far beyond the commutator relations of interchangeable observables (see e.g. quantum mechanics ) and states, among other things, that neighboring braids are not commutatable in a special way.)

The above algebraic definition is equivalent to the geometric one.

In particular, braid groups are a special case of the 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 ${\displaystyle B_{1}}$ ${\displaystyle B_{2}}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle B_{3}}$

${\displaystyle B_{3}=\langle \sigma _{1},\sigma _{2}\mid \sigma _{1}\sigma _{2}\sigma _{1}=\sigma _{2}\sigma _{1}\sigma _{2}\rangle }$

and is non-commutative.

Braid groups as figure class groups

The mapping class group of the circular disk with marked points is isomorphic to the braid group . ${\displaystyle n}$ ${\displaystyle B_{n}}$

Pure braid group

Each n-strand braid determines a permutation of the n-element set. The pure braid group is the core of the homomorphism so defined . ${\displaystyle B_{n}\rightarrow S_{n}}$

Applications

Mathematicians are particularly interested in the application in knot theory : by connecting the top of the braid to the bottom, you get a loop . Equivalent braids create equivalent tangles. On the other hand, each loop can be brought into the shape of a closed braid by isotopic transformation ( Alexander theorem ). Markow's sentence clarifies when two braids produce the same tangle (Andrei Andrejewitsch Markow, 1903–1979, son of Andrei Andrejewitsch Markow , 1856–1922).

literature

• Emil Artin : Theory of Braids. In: Treatises from the Mathematical Seminar of the University of Hamburg. 4, 1925, ISSN  0025-5858 , pp. 47-72.
• Joan Birman : Braids, Links, and Mapping Class Groups. Based on Lecture Notes by James Cannon. Princeton University Press, Princeton NJ 1975, ISBN 0-691-08149-2 ( Annals of Mathematics Studies 82).
• Moritz Epple : The emergence of the knot theory. Contexts and constructions of a modern mathematical theory. Vieweg, Braunschweig et al. 1999, ISBN 3-528-06787-X .
• Christian Kassel, Vladimir Turaev : Braid Groups. Springer, New York NY 2008, ISBN 978-0-387-33841-5 ( Graduate Texts in Mathematics 247).
• Bohdan I. Kurpita, Kunio Murasugi: A Study of Braids. Kluwer, Dordrecht et al. 1999, ISBN 0-7923-5767-1 ( Mathematics and its Applications 484).
• Vassily Manturov: Knot Theory. Routledge Chapman & Hall, Boca Raton FL et al. 2004, ISBN 0-415-31001-6 (online at GoogleBooks ).

Web links

Commons : Braid theory  - collection of images, videos and audio files

swell

1. ↑ See Epple's book on this.