Braid group - AbsoluteAstronomy.com

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the braid group on n strands, denoted by B_n, is a group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

which has an intuitive geometrical representation, and in a sense generalizes the symmetric group

Symmetric group

In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...

S_n. Here, n is a natural number

Natural number

In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...

; if n > 1, then B_n is an infinite group

Infinite group

In group theory, an area of mathematics, an infinite group is a group, of which the underlying set contains an infinite number of elements....

. Braid groups find applications in knot theory

Knot theory

In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...

, since any knot may be represented as the closure of certain braids.

Intuitive description

This introduction takes n to be 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 B₄. The above composition of braids is indeed a group

Group (mathematics)

operation. The neutral element is the braid consisting of four parallel horizontal strands, and the inverse

Inverse element

In abstract algebra, the idea of an inverse element generalises the concept of a negation, in relation to addition, and a reciprocal, in relation to multiplication. The intuition is of an element that can 'undo' the effect of combination with another given element...

of a braid consists of that braid which "undoes" whatever the first braid did. (The first two example braids above are inverses of each other.)

Formal treatment

To put the above informal discussion of braid groups on firm ground, one needs to use the homotopy

Homotopy

In topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions...

concept of algebraic topology

Algebraic topology

Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...

, defining braid groups as fundamental group

Fundamental group

In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...

s of a configuration space

Configuration space

- Configuration space in physics :In classical mechanics, the configuration space is the space of possible positions that a physical system may attain, possibly subject to external constraints...

. This is outlined in the article on braid theory

Braid theory

In topology, a branch of mathematics, braid theory is an abstract geometric theory studying the everyday braid concept, and some generalizations. 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...

.

Alternatively, one can eschew topology altogether and define the braid group purely algebraically via the braid relations, keeping the pictures in mind only to guide the intuition.

History

Braid groups were introduced explicitly by Emil Artin

Emil Artin

Emil Artin was an Austrian-American mathematician of Armenian descent.-Parents:Emil Artin was born in Vienna to parents Emma Maria, née Laura , a soubrette on the operetta stages of Austria and Germany, and Emil Hadochadus Maria Artin, Austrian-born of Armenian descent...

in 1925, although (as Wilhelm Magnus

Wilhelm Magnus

Wilhelm Magnus was a mathematician. He made important contributions in combinatorial group theory, Lie algebras, mathematical physics, elliptic functions, and the study of tessellations....

pointed out in 1974) they were already implicit in Adolf Hurwitz

Adolf Hurwitz

Adolf Hurwitz was a German mathematician.-Early life:He was born to a Jewish family in Hildesheim, former Kingdom of Hannover, now Lower Saxony, Germany, and died in Zürich, in Switzerland. Family records indicate that he had siblings and cousins, but their names have yet to be confirmed...

's work on monodromy

Monodromy

In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology and algebraic and differential geometry behave as they 'run round' a singularity. As the name implies, the fundamental meaning of monodromy comes from 'running round singly'...

(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

Braid theory

), an interpretation that was lost from view until it was rediscovered by Ralph Fox

Ralph Fox

Ralph Hartzler Fox was an American mathematician. As a professor at Princeton University, he taught and advised many of the contributors to the Golden Age of differential topology, and he played an important role in the modernization and main-streaming of knot theory.Ralph Fox attended Swarthmore...

and Lee Neuwirth in 1962.

Generators and relations

Consider the following three braids:


σ₁	σ₂	σ₃>

Every braid in B₄ can be written as a composition of a number of these braids and their inverses. In other words, these three braids generate

Generating set of a group

In abstract algebra, a generating set of a group is a subset that is not contained in any proper subgroup of the group. Equivalently, a generating set of a group is a subset such that every element of the group can be expressed as the combination of finitely many elements of the subset and their...

the group B₄. To see this, an arbitrary braid is scanned from left to right; whenever a crossing of strands i and i + 1 (counting from the top at the point of the crossing) is encountered, σ_i or σ_i⁻¹ is written down, depending on whether strand i moves under or over strand i + 1. Upon reaching the right hand end, the braid has been written as a product of the σ's and their inverses.

It is clear that
(i):σ₁σ₃ = σ₃σ₁,
while the following two relations are not quite as obvious:
(iia):σ₁σ₂σ₁ = σ₂σ₁σ₂,
(iib):σ₂σ₃σ₂ = σ₃σ₂σ₃
(these can be appreciated best by drawing the braid on a piece of paper). It can be shown that all other relations among the braids σ₁, σ₂ and σ₃ already follow from these relations and the group axioms.

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

Presentation of a group

In mathematics, one method of defining a group is by a presentation. One specifies a set S of generators so that every element of the group can be written as a product of powers of some of these generators, and a set R of relations among those generators...

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

The braid group B₁ is trivial
Trivial group
In mathematics, a trivial group is a group consisting of a single element. All such groups are isomorphic so one often speaks of the trivial group. The single element of the trivial group is the identity element so it usually denoted as such, 0, 1 or e depending on the context...

, B₂ is an infinite cyclic group
Cyclic group
In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...

Z, and B₃ is isomorphic to the knot group
Knot group
In mathematics, a knot is an embedding of a circle into 3-dimensional Euclidean space. The knot group of a knot K is defined as the fundamental group of the knot complement of K in R3,\pi_1....

of the trefoil knot
Trefoil knot
In topology, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining together the two loose ends of a common overhand knot, resulting in a knotted loop...

– in particular, it is an infinite non-abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

.

The n-strand braid group B_n embeds as a subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...

into the (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_n have infinite order
Order (group theory)
In group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....

: B_n is torsion-free.

Patrick Dehornoy
Patrick Dehornoy
Patrick Dehornoy is a mathematician at the University of Caen who works on set theory and algebra. He found one of the first applications of large cardinals to algebra by constructing a certain left-invariant total order, called the Dehornoy order, on the braid group.- References :*...

constructed a left-invariant linear order on B_n called the Dehornoy order
Dehornoy order
in mathematics, the Dehornoy order is a left-invariant total order on the braid group, found by .Dehornoy's original discovery of the order on the braid group used huge cardinals, but there are now several more elementary constructions of it.-Definition:...

.

For n ≥ 3, B_n contains a subgroup isomorphic to the free group
Free group
In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses...

on two generators.

There is a homomorphism
Group homomorphism
In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...

B_n → Z that maps every σ_i to 1. So for instance, the braid σ₂σ₃σ₁⁻¹σ₂σ₃ is mapped to 1 + 1 − 1 + 1 + 1 = 3.

Relation with symmetric group and the pure braid group

By forgetting how the strands twist and cross, every braid on n strands determines a permutation

Permutation

In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...

on n elements. This assignment is onto, compatible with composition, and therefore becomes a surjective group homomorphism

Group homomorphism

In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...

B_n → S_n from the braid group into the symmetric group

Symmetric group

. The image of the braid σ_i ∈ B_n is the transposition s_i = (i, i+1) ∈ S_n. 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

Coxeter group

In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...

of the symmetric group:

The kernel

Kernel (algebra)

In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. An important special case is the kernel of a matrix, also called the null space.The definition of kernel takes...

of the homomorphism B_n → S_n is the subgroup of B_n called the pure braid group on n strands and denoted P_n. In a pure braid, the beginning and the end of each strand are in the like positions. Pure braid groups fit into a short exact sequence

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

Semidirect product

In mathematics, specifically in the area of abstract algebra known as group theory, a semidirect product is a particular way in which a group can be put together from two subgroups, one of which is a normal subgroup. A semidirect product is a generalization of a direct product...

of free groups.

Relation between B₃ and the modular group

The braid group B₃ is the universal central extension of the modular group

Modular group

In mathematics, the modular group Γ is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics...

PSL(2,Z), with these sitting as lattices inside the (topological) universal covering group

Further, the modular group has trivial center, and thus the modular group is isomorphic to the quotient group

Quotient group

In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...

modulo its center

Center (group theory)

In abstract algebra, the center of a group G, denoted Z,The notation Z is from German Zentrum, meaning "center". is the set of elements that commute with every element of G. In set-builder notation,...

;
equivalently, to the group of inner automorphism

Inner automorphism

In abstract algebra an inner automorphism is a functionwhich, informally, involves a certain operation being applied, then another one performed, and then the initial operation being reversed...

s of

.

A construction is given below.

Define

and

. From the braid relations it follows that

. Denoting this latter product as

, one may verify from the braid relations that

implying that

is in the center of B₃. The subgroup

Subgroup

In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...

of B₃ generated by

is therefore a normal subgroup

Normal subgroup

In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....

. Since it is normal, one may take the quotient group

Quotient group

In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...

; this quotient group is isomorphic to the modular group:

This isomorphism can be given an explicit form. The coset

Coset

In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, thenA coset is a left or right coset of some subgroup in G...

and

map to

where

and

are the standard left and right moves on the Stern-Brocot tree

Stern-Brocot tree

In number theory, the Stern–Brocot tree is an infinite complete binary tree in which the vertices correspond precisely to the positive rational numbers, whose values are ordered from left to right as in a search tree....

; it is well known that these moves generate the modular group.

Alternately, one common presentation

Presentation of a group

for the modular group is

where

Mapping a to v and b to p yields a surjective group homomorphism from B₃ to PSL(2,Z).

The center of B₃ is equal to

, a consequence of the facts that c is in the center, the modular group has trivial center, and the above surjective homomorphism has kernel

Kernel (algebra)

Relationship to the mapping class group and classification of braids

The braid group B_n can be shown to be isomorphic to the mapping class group

Mapping class group

In mathematics, in the sub-field of geometric topology, the mapping class groupis an important algebraic invariant of a topological space. Briefly, the mapping class group is a discrete group of 'symmetries' of the space.-Motivation:...

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 homeomorphism 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

Nielsen-Thurston classification

In mathematics, Thurston's classification theorem characterizes homeomorphisms of a compact surface. William Thurston's theorem completes the work initiated by ....

Connection to knot theory and computational aspects

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

Link (knot theory)

In mathematics, a link is a collection of knots which do not intersect, but which may be linked together. A knot can be described as a link with one component. Links and knots are studied in a branch of mathematics called knot theory...

, and sometimes a knot

Knot (mathematics)

In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations . A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed—there are no ends to tie or untie on a...

. Alexander's theorem in braid theory

Braid theory

states that the converse is true as well: every knot

Knot (mathematics)

and every link

Link (knot theory)

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.

The word problem

Word problem for groups

In mathematics, especially in the area of abstract algebra known as combinatorial group theory, the word problem for a finitely generated group G is the algorithmic problem of deciding whether two words in the generators represent the same element...

for the braid relations is efficiently solvable and there exists a normal form

Normal form

Normal form may refer to:* Normal form * Normal form * Normal form * Normal form In formal language theory:* Beta normal form* Chomsky normal form* Greibach normal form* Kuroda normal form...

for elements of B_n in terms of the generators σ₁,...,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

GAP computer algebra system

GAP is a computer algebra system for computational discrete algebra with particular emphasis on computational group theory.-History:...

can carry out computations in B_n 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.

Since there are nevertheless several hard computational problems about braid groups, applications in cryptography

Cryptography

Cryptography is the practice and study of techniques for secure communication in the presence of third parties...

have been suggested.

Actions of braid groups

In analogy with the action of the symmetric group by permutations, in various mathematical settings there exists a natural action the braid group on n-tuples of objects or on the n-folded tensor product

Tensor product

In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...

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

Identity element

In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...

of G. Then B_n acts

Group action

In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...

on X in the following fashion:

Thus the elements x_i and x_i+1 exchange places and, in addition, x_i is twisted by the inner automorphism

Inner automorphism

In abstract algebra an inner automorphism is a functionwhich, informally, involves a certain operation being applied, then another one performed, and then the initial operation being reversed...

corresponding to 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_n on X. As another example, a braided monoidal category is a monoidal category

Monoidal category

In mathematics, a monoidal category is a category C equipped with a bifunctorwhich is associative, up to a natural isomorphism, and an object I which is both a left and right identity for ⊗, again up to a natural isomorphism...

with a braid group action. Such structures play an important role in modern mathematical physics

Mathematical physics

Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...

and lead to quantum knot invariant

Knot invariant

In the mathematical field of knot theory, a knot invariant is a quantity defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some invariants are indeed numbers, but invariants can range from the...

Representations

Elements of the braid group B_n can be represented more concretely by matrices. One classical such representation

Group representation

In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...

is Burau representation

Burau representation

In mathematics the Burau representation is a representation of the braid groups, named after and originally studied by the German mathematician Werner Burau during the 1930s...

, where the matrix entries are single variable Laurent polynomials. It had been a long-standing question whether Burau representation was faithful

Faithful representation

In mathematics, especially in the area of abstract algebra known as representation theory, a faithful representation ρ of a group G on a vector space V is a linear representation in which different elements g of G are represented by distinct linear mappings ρ.In more abstract language, this means...

, 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

Ruth Lawrence

Ruth Elke Lawrence-Naimark is an Associate Professor of mathematics at the Einstein Institute of Mathematics, Hebrew University of Jerusalem, and a researcher in knot theory and algebraic topology. Outside academia, she is best known for being a child prodigy in mathematics.- Youth :Ruth Lawrence...

described a family of more general "Lawrence representations" depending on several parameters. Around 2001 Stephen Bigelow and Daan Krammer independently proved that all braid groups are linear. Their work used the Lawrence–Krammer representation

Lawrence–Krammer representation

In mathematics the Lawrence–Krammer representation is a representation of the braid groups. It fits into a family of representations called the Lawrence representations...

of dimension n(n−1)/2 depending on the variables q and t. By suitably specializing these variables, the braid group B_n may be realized as a subgroup of the general linear group

General linear group

In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...

over the complex numbers.

Infinitely generated braid groups

There are many ways to generalize this notion to an infinite number of strands. The simplest way is take the direct limit

Direct limit

In mathematics, a direct limit is a colimit of a "directed family of objects". We will first give the definition for algebraic structures like groups and modules, and then the general definition which can be used in any category.- Algebraic objects :In this section objects are understood to be...

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

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

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

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

Inverse limit

In mathematics, the inverse limit is a construction which allows one to "glue together" several related objects, the precise manner of the gluing process being specified by morphisms between the objects...

of finite pure braid groups

and to the fundamental group

Fundamental group

of the Hilbert cube

Hilbert cube

In mathematics, the Hilbert cube, named after David Hilbert, is a topological space that provides an instructive example of some ideas in topology...

minus the set

External links

CRAG: CRyptography and Groups at Algebraic Cryptography Center Contains extensive library for computations with Braid Groups
P. Fabel, Completing Artin's braid group on infinitely many strands, Journal of Knot Theory and its Ramifications, Vol. 14, No. 8 (2005) 979–991
P. Fabel, The mapping class group of a disk with infinitely many holes, Journal of Knot Theory and its Ramifications, Vol. 15, No. 1 (2006) 21–29
Braid Theory, Encyclopaedia of Mathematics, Springer 2002
Stephen Bigelow's exploration of B5 Java applet.
Cryptography and Braid Groups page - Helger Lipmaa
Braid group: (List of Authority Articles on arxiv.org:

The source of this article is wikipedia, the free encyclopedia. The text of this article is licensed under the GFDL.

Intuitive description

Formal treatment

History

Generators and relations

Further properties

Relation with symmetric group and the pure braid group

Relation between B3 and the modular group

Relationship to the mapping class group and classification of braids

Connection to knot theory and computational aspects

Actions of braid groups

Representations

Infinitely generated braid groups

Further reading

External links

Relation between B₃ and the modular group