# Braid group

It has been suggested that Braid theory be merged into this article. (Discuss) Proposed since September 2017. |

In mathematics, the **braid group on n strands** (denoted

*B*), also known as the

_{n}**Artin braid group**,

^{[1]}is the group whose elements are equivalence classes of n-braids (e.g. under ambient isotopy), and whose group operation is composition of braids (see § Introduction). Example applications of braid groups include knot theory, where any knot may be represented as the closure of certain braids (a result known as Alexander's theorem); in mathematical physics where Artin's canonical presentation of the braid group corresponds to the Yang–Baxter equation (see § Basic properties); and in monodromy invariants of algebraic geometry.

^{[2]}

## Contents

## Introduction[edit]

### Intuitive description[edit]

In this introduction let *n* = 4; the generalization to other values of *n* will be straightforward. Consider two sets of four items lying on a table, with the items in each set being arranged in a vertical line, and such that one set sits next to the other. (In the illustrations below, these are the black dots.) Using four strands, each item of the first set is connected with an item of the second set so that a one-to-one correspondence results. Such a connection is called a *braid*. Often some strands will have to pass over or under others, and this is crucial: the following two connections are *different* braids:

is different from |

On the other hand, two such connections which can be made to look the same by "pulling the strands" are considered *the same* braid:

is the same as |

All strands are required to move from left to right; knots like the following are *not* considered braids:

is not a braid |

Any two braids can be *composed* by drawing the first next to the second, identifying the four items in the middle, and connecting corresponding strands:

composed with | yields |

Another example:

composed with | yields |

The composition of the braids σ and τ is written as στ.

The set of all braids on four strands is denoted by *B*_{4}. The above composition of braids is indeed a group operation. The identity element is the braid consisting of four parallel horizontal strands, and the inverse of a braid consists of that braid which "undoes" whatever the first braid did, which is obtained by flipping a diagram such as the ones above across a vertical line going through its centre. (The first two example braids above are inverses of each other.)

### Formal treatment[edit]

To put the above informal discussion of braid groups on firm ground, one needs to use the homotopy concept of algebraic topology, defining braid groups as fundamental groups of a configuration space. Alternatively, one can define the braid group purely algebraically via the braid relations, keeping the pictures in mind only to guide the intuition.

### History[edit]

Braid groups were introduced explicitly by Emil Artin in 1925, although (as Wilhelm Magnus pointed out in 1974^{[3]}) they were already implicit in Adolf Hurwitz's work on monodromy from 1891. In fact, as Magnus says, Hurwitz gave the interpretation of a braid group as the fundamental group of a configuration space (cf. braid theory), an interpretation that was lost from view until it was rediscovered by Ralph Fox and Lee Neuwirth in 1962.^{[4]}

## Basic properties[edit]

### Generators and relations[edit]

Consider the following three braids:

_{1} | _{2} | _{3} |

Every braid in can be written as a composition of a number of these braids and their inverses. In other words, these three braids generate the group . To see this, an arbitrary braid is scanned from left to right for crossings; beginning at the top, whenever a crossing of strands and is encountered, or is written down, depending on whether strand moves under or over strand . Upon reaching the right end, the braid has been written as a product of the σ's and their inverses.

It is clear that

- (i) σ
_{1}σ_{3}= σ_{3}σ_{1},

- (i) σ

while the following two relations are not quite as obvious:

- (iia) σ
_{1}σ_{2}σ_{1}= σ_{2}σ_{1}σ_{2}, - (iib) σ
_{2}σ_{3}σ_{2}= σ_{3}σ_{2}σ_{3}

- (iia) σ

(these relations can be appreciated best by drawing the braid on a piece of paper). It can be shown that all other relations among the braids , and already follow from these relations and the group axioms.

Generalising this example to strands, the group can be abstractly defined via the following presentation:

where in the first group of relations 1 ≤ *i* ≤ *n*−2 and in the second group of relations, |*i* − *j*| ≥ 2. This presentation leads to generalisations of braid groups called Artin groups. The cubic relations, known as the **braid relations**, play an important role in the theory of Yang–Baxter equation.

### Further properties[edit]

- The braid group
*B*_{1}is trivial,*B*_{2}is an infinite cyclic group**Z**, and*B*_{3}is isomorphic to the knot group of the trefoil knot – in particular, it is an infinite non-abelian group. - The
*n*-strand braid group*B*embeds as a subgroup into the (_{n}*n*+1)-strand braid group*B*_{n+1}by adding an extra strand that does not cross any of the first*n*strands. The increasing union of the braid groups with all*n*≥ 1 is the**infinite braid group***B*_{∞}. - All non-identity elements of
*B*have infinite order; i.e.,_{n}*B*is torsion-free._{n} - There is a left-invariant linear order on
*B*called the Dehornoy order._{n} - For
*n*≥ 3,*B*contains a subgroup isomorphic to the free group on two generators._{n} - There is a homomorphism
*B*→_{n}**Z**defined by σ_{i}↦ 1. So for instance, the braid σ_{2}σ_{3}σ_{1}^{−1}σ_{2}σ_{3}is mapped to 1 + 1 − 1 + 1 + 1 = 3. This map corresponds to the abelianization of the braid group.

## Interactions[edit]

### Relation with symmetric group and the pure braid group[edit]

By forgetting how the strands twist and cross, every braid on *n* strands determines a permutation on *n* elements. This assignment is onto, compatible with composition, and therefore becomes a surjective group homomorphism *B _{n}* →

*S*from the braid group into the symmetric group. The image of the braid σ

_{n}_{i}∈

*B*is the transposition

_{n}*s*

_{i}= (

*i*,

*i*+1) ∈

*S*. These transpositions generate the symmetric group, satisfy the braid group relations, and have order 2. This transforms the Artin presentation of the braid group into the Coxeter presentation of the symmetric group:

_{n}The kernel of the homomorphism *B _{n}* →

*S*is the subgroup of

_{n}*B*called the

_{n}**pure braid group on**and denoted

*n*strands*P*. In a pure braid, the beginning and the end of each strand are in the same position. Pure braid groups fit into a short exact sequence

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

### Relation between *B*_{3} and the modular group[edit]

The braid group *B*_{3} is the universal central extension of the modular group PSL(2, **Z**), with these sitting as lattices inside the (topological) universal covering group

Furthermore, the modular group has trivial center, and thus the modular group is isomorphic to the quotient group of *B*_{3} modulo its center, *Z*(*B*_{3}), and equivalently, to the group of inner automorphisms of *B*_{3}.

Here is a construction of this isomorphism. Define

From the braid relations it follows that *a*^{2} = *b*^{3}. Denoting this latter product as *c*, one may verify from the braid relations that

implying that *c* is in the center of *B*_{3}. Let *C* denote the subgroup of *B*_{3} generated by *c*, since *C* ⊂ *Z*(*B*_{3}), it is a normal subgroup and one may take the quotient group *B*_{3}/*C*. We claim *B*_{3}/*C* ≅ PSL(2, **Z**); this isomorphism can be given an explicit form. The cosets σ_{1}*C* and σ_{2}*C* map to

where *L* and *R* are the standard left and right moves on the Stern–Brocot tree; it is well known that these moves generate the modular group.

Alternately, one common presentation for the modular group is

where

Mapping *a* to *v* and *b* to *p* yields a surjective group homomorphism *B*_{3} → PSL(2, **Z**).

The center of *B*_{3} is equal to *C*, a consequence of the facts that *c* is in the center, the modular group has trivial center, and the above surjective homomorphism has kernel *C*.

### Relationship to the mapping class group and classification of braids[edit]

The braid group *B _{n}* can be shown to be isomorphic to the mapping class group of a punctured disk with

*n*punctures. This is most easily visualized by imagining each puncture as being connected by a string to the boundary of the disk; each mapping homomorphism that permutes two of the punctures can then be seen to be a homotopy of the strings, that is, a braiding of these strings.

Via this mapping class group interpretation of braids, each braid may be classified as periodic, reducible or pseudo-Anosov.

### Connection to knot theory[edit]

If a braid is given and one connects the first left-hand item to the first right-hand item using a new string, the second left-hand item to the second right-hand item etc. (without creating any braids in the new strings), one obtains a link, and sometimes a knot. Alexander's theorem in braid theory states that the converse is true as well: every knot and every link arises in this fashion from at least one braid; such a braid can be obtained by cutting the link. Since braids can be concretely given as words in the generators σ_{i}, this is often the preferred method of entering knots into computer programs.

### Computational aspects[edit]

The word problem for the braid relations is efficiently solvable and there exists a normal form for elements of *B _{n}* in terms of the generators σ

_{1}, ..., σ

_{n−1}. (In essence, computing the normal form of a braid is the algebraic analogue of "pulling the strands" as illustrated in our second set of images above.) The free GAP computer algebra system can carry out computations in

*B*if the elements are given in terms of these generators. 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.

_{n}Since there are nevertheless several hard computational problems about braid groups, applications in cryptography have been suggested.

## Actions[edit]

In analogy with the action of the symmetric group by permutations, in various mathematical settings there exists a natural action of the braid group on *n*-tuples of objects or on the *n*-folded tensor product that involves some "twists". Consider an arbitrary group *G* and let *X* be the set of all *n*-tuples of elements of *G* whose product is the identity element of *G*. Then *B _{n}* acts on

*X*in the following fashion:

Thus the elements *x _{i}* and

*x*

_{i+1}exchange places and, in addition,

*x*is twisted by the inner automorphism corresponding to

_{i}*x*

_{i+1}— this ensures that the product of the components of

*x*remains the identity element. It may be checked that the braid group relations are satisfied and this formula indeed defines a group action of

*B*on

_{n}*X*. As another example, a braided monoidal category is a monoidal category with a braid group action. Such structures play an important role in modern mathematical physics and lead to quantum knot invariants.

### Representations[edit]

Elements of the braid group *B _{n}* can be represented more concretely by matrices. One classical such representation is Burau representation, where the matrix entries are single variable Laurent polynomials. It had been a long-standing question whether Burau representation was faithful, but the answer turned out to be negative for

*n*≥ 5. More generally, it was a major open problem whether braid groups were linear. In 1990, Ruth Lawrence described a family of more general "Lawrence representations" depending on several parameters. In 1996, C. Nayak and Frank Wilczek posited that in analogy to projective representations of SO(3), the projective representations of the braid group have a physical meaning for certain quasiparticles in the fractional quantum hall effect. Around 2001 Stephen Bigelow and Daan Krammer independently proved that all braid groups are linear. Their work used the Lawrence–Krammer representation of dimension

*n*(

*n*−1)/2 depending on the variables

*q*and

*t*. By suitably specializing these variables, the braid group

*B*may be realized as a subgroup of the general linear group over the complex numbers.

_{n}## Infinitely generated braid groups[edit]

There are many ways to generalize this notion to an infinite number of strands. The simplest way is to take the direct limit of braid groups, where the attaching maps send the generators of to the first generators of (i.e., by attaching a trivial strand). Fabel has shown that there are two topologies that can be imposed on the resulting group each of whose completion yields a different group. 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. An interesting fact is that the pure braid group in this group is isomorphic to both the inverse limit of finite pure braid groups and to the fundamental group of the Hilbert cube minus the set

## Cohomology[edit]

The cohomology of a group is defined as the cohomology of the corresponding Eilenberg–MacLane classifying space, , which is a CW complex uniquely determined by up to homotopy. A classifying space for the braid group is the `n`^{th} unordered configuration space of , that is, the set of distinct unordered points in the plane:^{[5]}

- .

So by definition

- .

The calculations for coefficients in can be found in Fuchs 1970.^{[6]}

Similarly, a classifying space for the pure braid group is , the `n`^{th} *ordered* configuration space of . In 1968 Vladimir Arnold showed that the integral cohomology of the pure braid group is the quotient of the exterior algebra generated by the collection of degree-one classes , subject to the relations^{[7]}

## See also[edit]

## Notes[edit]

**^**Weisstein, Eric. "Braid Group".*Wolfram Mathworld*.**^**Cohen, Daniel; Suciu, Alexander (1997). "The Braid Monodromy of Plane Algebraic Curves and Hyperplane Arrangements".*Commentarii Mathematici Helvetici 72*.**72**(2): 285–315. arXiv:alg-geom/9608001. doi:10.1007/s000140050017.**^**Magnus, Wilhelm (1974). "Braid groups: A survey".*Proceedings of the Second International Conference on the Theory of Groups*. Lecture Notes in Mathematics.**372**. Springer. pp. 463–487. ISBN 978-3-540-06845-7.**^**Fox, Ralph; Neuwirth, Lee (1962). "The braid groups".*Mathematica Scandinavica*.**10**: 119–126. doi:10.7146/math.scand.a-10518. MR 0150755.**^**Ghrist, Robert (2009-12-01).*Braids*. Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore. Volume 19. WORLD SCIENTIFIC. pp. 263–304. doi:10.1142/9789814291415_0004. ISBN 9789814291408.**^**Fuks, Dmitry B. (1970).*Cohomology of the braid group mod 2*.*Functional Analysis and its Applications*.**4**. pp. 143–151. doi:10.1007/BF01094491. MR 0274463.**^**Arnol'd, Vladimir. "The cohomology ring of the colored braid group" (PDF).

## References[edit]

- Deligne, Pierre (1972), "Les immeubles des groupes de tresses généralisés",
*Inventiones Mathematicae*,**17**(4): 273–302, Bibcode:1972InMat..17..273D, doi:10.1007/BF01406236, ISSN 0020-9910, MR 0422673

## Further reading[edit]

- Birman, Joan; Brendle, Tara E. (26 February 2005),
*Braids: A Survey*, arXiv:math.GT/0409205. In Menasco & Thistlethwaite 2005 - Carlucci, Lorenzo; Dehornoy, Patrick; Weiermann, Andreas (2011), "Unprovability results involving braids",
*Proceedings of the London Mathematical Society*, 3,**102**(1): 159–192, arXiv:0711.3785, doi:10.1112/plms/pdq016, MR 2747726 - Kassel, Christian; Turaev, Vladimir (2008),
*Braid Groups*, Springer, ISBN 0-387-33841-1 - Menasco, William; Thistlethwaite, Morwen, eds. (2005),
*Handbook of Knot Theory*, Elsevier, ISBN 0-444-51452-X

## External links[edit]

- "Braid group".
*PlanetMath*. - CRAG: CRyptography and Groups at Algebraic Cryptography Center Contains extensive library for computations with Braid Groups
- [1] Visual Group Theory, Lecture 1.3: Groups in science, art, and mathematics
- Fabel, P. (2005), "Completing Artin's braid group on infinitely many strands" (PDF),
*Journal of Knot Theory and its Ramifications*,**14**(8): 979–991, arXiv:math/0201303, doi:10.1142/S0218216505004196, archived from the original (PDF) on 9 January 2006 - Fabel, P. (2006), "The mapping class group of a disk with infinitely many holes" (PDF),
*Journal of Knot Theory and its Ramifications*,**15**(1): 21–29, arXiv:math/0303042, doi:10.1142/S0218216506004324, archived from the original (PDF) on 29 January 2005 - Chernavskii, A.V. (2001) [1994], "Braid theory", in Hazewinkel, Michiel,
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 - Stephen Bigelow's exploration of B5 Java applet.
- Nayak, C.; Wilczek, F. (1996), "$2n$ Quasihole States Realize $2^{n-1}$-Dimensional Spinor Braiding Statistics in Paired Quantum Hall States",
*Nuclear Physics B*,**479**: 529–553, arXiv:cond-mat/9605145, Bibcode:1996NuPhB.479..529N, doi:10.1016/0550-3213(96)00430-0 — connection of projective braid group representations to the fractional quantum Hall effect - Presentation for FradkinFest by C. V. Nayak [2]
^{[permanent dead link]} - Read, N. (2002),
*Nonabelian braid statistics versus projective permutation statistics*, arXiv:hep-th/0201240, Bibcode:2003JMP....44..558R, doi:10.1063/1.1530369 — criticism of the reality of Wilczek-Nayak representation - Lipmaa, Helger,
*Cryptography and Braid Groups page*, archived from the original on 3 August 2009 - Braid group: List of Authority Articles on arxiv.org.