Braid group

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In mathematics, the braid group on n strands, denoted by Bn, is a certain group which has an intuitive geometrical representation, and in a sense generalizes the symmetric group Sn. Here, n is a natural number; if n > 1, then Bn is an infinite group. Braid groups find applications in knot theory, since any knot may be represented as the closure of certain braids.

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[edit] 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:

The braid sigma_1^(-1)    is different from    The braid sigma_1

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

The braid sigma_1^(-1)     is the same as    Another representation of sigma_1^(-1)

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

Not a braid    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:

image:braid_s3.png     composed with     image:braid_s2.png     yields     image:braid_s3s2.png

Another example:

image:braid_s1_inv_s3_inv.png     composed with     image:braid_s1_s3_inv.png     yields     image:braid_s3_inv_squared.png

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 operation. The neutral 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. (The first two example braids above are inverses of each other.)

[edit] Generators and relations

Consider the following three braids:

   image:braid_s1.png       image:braid_s2.png       image:braid_s3.png   
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 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, σi or σi−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

σ1σ3 = σ3σ1,

while the following two relations are not quite as obvious:

σ1σ2σ1 = σ2σ1σ2
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 Bn can be abstractly defined via the following presentation:

  • generators σ1,...,σn−1
  • relations (known as the braid relations):
    • σi σj = σj σi whenever |i − j| ≥ 2 ;
    • σi σi+1 σi = σi+1 σi σi+1 for i = 1,..., n − 2 (sometimes called the Yang-Baxter equation)

[edit] Some properties

The groups B0 and B1 are trivial; B2 is already infinite and isomorphic to the infinite cyclic group Z. B3 is a quite complicated non-abelian infinite group; in fact, B3 is isomorphic to the knot group of the trefoil.

In general, Bn is a subgroup of Bn + 1: it can be viewed as consisting of all those braids on n + 1 strands in which the bottom strand is horizontal and does not cross nor is crossed by any other strand.

So in particular, Bn is abelian if and only if n ≤ 2.

There is a useful notion of "length" for the elements of the braid group, given by the group homomorphism Bn → Z that maps every σi to 1. So for instance, the length of the braid σ2σ3σ1−1σ2σ3 is 1 + 1 − 1 + 1 + 1 = 3. This notion gives rise, for example, to the subgroup of Bn consisting of all even-length braids.

Bn is torsion-free.

[edit] Relation to the symmetric group, group actions

Every braid on n strands basically consists of a one-to-one correspondence between two sets of n items, and some topological information about how the strands establish this correspondence. Without this topological information every braid yields a one-to-one correspondence of n items; these are precisely the elements of the symmetric group Sn. This assignment is in fact a surjective group homomorphism Bn → Sn.

The kernel of this group homomorphism is called the pure braid group on n strands; it consists of those braids which connect the ith item of the left set to the ith item of the right set, for all i.

The symmetric group Sn has a very similar presentation to the one given above for the braid group: taking the braid relations and adding the relations

σi2 = 1 for i = 1, ..., n − 1

yields a presentation for Sn (the σi can then be thought of as transpositions of two neighboring elements).

In situations where n items are being permuted "up to a twist", there is often an underlying group action of the braid group Bn. As a prototypical example, consider an arbitrary group G and the set X of all n-tuples of elements of G whose product is 1, the identity element of G. Then Bn operates on X in the following natural fashion: given a tuple x = (x1, ..., xn) in X define σi.x = (x1, ..., xi−1, xi+1, xi+1−1xixi+1, xi+2, ..., xn), so xi and xi+1 exchange places, but xi is in addition "twisted" by the inner automorphism corresponding to xi+1; this twist ensures that the product of the components of σi.x is the same as that of the components of x, namely 1. This operation satisfies the braid relations and thus defines a group action of Bn on X.

[edit] Relation between B3 and the modular group

There is a surjective homomorphism from B3 onto the modular group PSL_2(\mathbb{Z}) with kernel equal to the center of B3; a construction is given below.

Define a = σ1σ2σ1 and b = σ1σ2. From the braid relations it follows that a2 = b3. Denoting this latter product as c = a2 = b3, one may verify from the braid relations that

\sigma_1 c \sigma_1^{-1} = \sigma_2 c \sigma_2^{-1}=c

implying that c is in the center of B3. The subgroup \langle c\rangle of B3 generated by c is therefore a normal subgroup. Since it is normal, one may take the quotient group; this quotient group is isomorphic to the modular group:

PSL_2(\mathbb{Z}) \simeq B_3/\langle c\rangle.

This isomorphism can be given an explicit form. The cosets [σ1] of σ1 and [σ2] of σ2 map to

[\sigma_1] \mapsto R=\begin{bmatrix}1 & 1 \\ 0 & 1 \end{bmatrix} \qquad [\sigma_2] \mapsto L^{-1}=\begin{bmatrix}1 & 0 \\ -1 & 1 \end{bmatrix}

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

\langle v,p\, |\, v^2=p^3=1\rangle

where

a = \sigma_1 \sigma_2 \sigma_1 \mapsto v=\begin{bmatrix}0 & 1 \\ -1 & 0 \end{bmatrix}

and

b = \sigma_1 \sigma_2 \mapsto p=\begin{bmatrix}0 & 1 \\ -1 & 1 \end{bmatrix}

with

c = a^2 = b^3 \mapsto \begin{bmatrix}-1 & 0 \\ 0 & -1 \end{bmatrix}

the latter being the identity element of PSL_2(\mathbb{Z}).

The center of B3 is equal to \langle c\rangle, a consequence of the facts that c is in the center, the modular group has trivial center, and the above surjective homomorphism has kernel \langle c\rangle.

[edit] Relationship to the mapping class group and the monodromy

The braid group Bn can be shown to be 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 homeomorphism that permutes of two of the punctures can then be seen to be a homotopy of the strings, that is, a braiding of these strings.

The braid group may be mapped onto the monodromy of an analytic function. This may be visualized by considering a disk with n-1 punctures, each puncture corresponding to a pole of the analytic function. The monodromy can then be visualized by taking each of the punctures to be a straight line perpendicular to the disk, and the monodromy path as a string, anchored at a point, that winds around each of the punctures, returning to its original starting point.

[edit] 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, 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 σi, this is often the preferred method of entering knots into computer programs.

The word problem for the braid relations is efficiently solvable and there exists a normal form for elements of Bn in terms of the generators σ1,...,σn−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 Bn 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.

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

[edit] 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 of braid groups, where the attaching maps f:B_n\to B_{n+1} send the n − 1 generators of Bn to the first n − 1 generators of Bn + 1 (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 (0,1 / n) and the set of all braids--where a braid is defined to be a collection of paths from the points (0,1 / n,0) to the points (0,1 / n,1) 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 Pn and to the fundamental group of the Hilbert cube minus the set \{(x_i)_{i\in \Bbb{N}} \mid x_i=x_j\text{ for some }i\ne j\}.

[edit] Formal treatment

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. This is outlined in the article on braid theory.

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.

[edit] History

Braid groups were introduced explicitly by Emil Artin in 1925, although (as Wilhelm Magnus pointed out in 1974[1]) they were already implicit in Adolf Hurwitz's work on monodromy (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.

[edit] References

  1. ^ Wilhelm Magnus. Braid groups: A survey. In Lecture Notes in Mathematics, volume 372, pages 463-487. Springer, 1974.

[edit] External links