Braid group

In mathematics, the braid group on "n" strands, denoted by "B"_"n", is a certain group which has an intuitive geometrical representation, and in a sense generalizes the symmetric group "S"_"n". Here, "n" is a natural number; if "n" > 1, then "B"_"n" is an infinite group. Braid groups find applications in knot theory, 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::

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::

Every braid in "B"₄ can be written as a composition of a number of these braids and their inverses. In other words, these three braids generate the group "B"₄. 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: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 "B"_"n" can be abstractly defined via the following presentation:
* generators σ₁,...,σ_"n"−1
* relations (known as the "braid or Artin relations"):
** σ_"i" σ_"j" = σ_"j" σ_"i" whenever |"i" − "j"| ≥ 2 ;
** σ_"i" σ_"i"+1 σ_"i" = σ_"i"+1 σ_"i" σ_"i"+1 for "i" = 1,..., "n" − 2 (sometimes called the Yang-Baxter equation)

Some properties

The groups "B"₀ and "B"₁ are trivial; "B"₂ is an infinite cyclic group. "B"₃ is a non-abelian infinite group; in fact, "B"₃ is isomorphic to the knot group of the trefoil.

Provided $n\; >\; 2$, $B\_n$ contains a free group on two generators, and so it is not abelian.

"B"_"n" is a subgroup of "B"_{"n" + 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. The formal union of all the braid groups is sometimes called the "infinite braid group".

There is a useful notion of "length" for the elements of the braid group, given by the group homomorphism "B"_"n" → Z that maps every σ_"i" 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 "B"_"n" consisting of all even-length braids.

"B"_"n" is torsion-free.

Via the mapping-class group interpretation of braids, all braids have a classification as either periodic, reducible or pseudo-anosov.

"B"_"n" is known to be a subgroup of the unitary group $U\{n\; choose\; 2\}$. The embedding is given by the Lawrence-Krammer representation where the variables $q,t$ are specialized to suitable algebraically-independent unit complex numbers.

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 "S"_"n". This assignment is in fact a surjective group homomorphism "B"_"n" → "S"_"n".

The kernel of this group homomorphism is called the pure braid group on "n" strands $P\_n$; it consists of those braids which connect the "i"-th item of the left set to the "i"-th item of the right set, for all "i". There are split group extensions $F\_\{n-1\}\; o\; P\_n\; o\; P\_\{n-1\}$ ie: pure braid groups are iterated semi-direct products of free groups.

The symmetric group "S"_"n" has a very similar presentation to the one given above for the braid group: taking the braid relations and adding the relations:σ_"i"² = 1 for "i" = 1, ..., "n" − 1yields a presentation for "S"_"n" (the σ_"i" 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 "B"_"n". 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 "B"_"n" operates on "X" in the following natural fashion: given a tuple "x" = ("x"₁, ..., "x"_"n") in "X" define σ_"i"."x" = ("x"₁, ..., "x"_"i"−1, "x"_"i"+1, "x"_"i"+1⁻¹"x"_"i""x"_"i"+1, "x"_"i"+2, ..., "x"_"n"), so "x"_"i" and "x"_"i"+1 exchange places, but "x"_"i" is in addition "twisted" by the inner automorphism corresponding to "x"_"i"+1; this twist ensures that the product of the components of σ_"i"."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 "B"_"n" on "X".

Relation between "B"₃ and the modular group

There is a surjective homomorphism from "B"₃ onto the modular group $PSL\_2(mathbb\{Z\})$ with kernel equal to the center of "B"₃; a construction is given below.

Define $a\; =\; sigma\_1\; sigma\_2\; sigma\_1$ and $b\; =\; sigma\_1\; sigma\_2$. From the braid relations it follows that $a^2=b^3$. Denoting this latter product as $c=a^2=b^3$, 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 "B"₃. The subgroup $langle\; c\; angle$ of "B"₃ 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:

:$PSL\_2(mathbb\{Z\})\; simeq\; B\_3/langle\; c\; angle.$

This isomorphism can be given an explicit form. The cosets $[sigma\_1]$ of $sigma\_1$ and $[sigma\_2]$ of $sigma\_2$ 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

:$langle\; v,p,\; |,\; v^2=p^3=1\; angle$

where

:

and

:

with:

the latter being the identity element of $PSL\_2(mathbb\{Z\})$.

The center of "B"₃ is equal to $langle\; c\; angle$, a consequence of the facts that "c" is in the center, the modular group has trivial center, and the above surjective homomorphism has kernel $langle\; c\; angle$.

Relationship to the mapping class group and the monodromy

The braid group "B"_n 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 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.

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 σ_"i", 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 "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 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 have been suggested.

Representations

Frequently referenced representations of the braid groups include the Burau representation, the Lawrence-Krammer representation and the Lawrence representations.

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 $f:B\_n\; o\; B\_\{n+1\}$ send the $n-1$ generators of $B\_n$ to the first $n-1$ generators of $B\_\{n+1\}$ (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 $P\_n$ and to the fundamental group of the Hilbert cube minus the set $\{(x\_i)\_\{iin\; Bbb\{N\; mid\; x\_i=x\_j\; ext\{\; for\; some\; \}i\; e\; j\}$.

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.

History

Braid groups were introduced explicitly by Emil Artin in 1925, although (as Wilhelm Magnus pointed out in 1974 [Wilhelm Magnus. Braid groups: A survey. In "Lecture Notes in Mathematics", volume 372, pages 463-487. Springer, 1974.] ) 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.

References

*Citation | last1=Deligne | first1=Pierre | author1-link=Pierre Deligne | title=Les immeubles des groupes de tresses généralisés | doi=10.1007/BF01406236 | id=MathSciNet | id = 0422673 | year=1972 | journal=Inventiones Mathematicae | issn=0020-9910 | volume=17 | pages=273–302

External links

*planetmath reference|id=4604|title=Braid group
* [http://www.acc.stevens.edu/downloads.php CRAG: CRyptography and Groups] at [http://www.acc.stevens.edu Algebraic Cryptography Center] Contains extensive library for computations with Braid Groups
*P. Fabel, [http://www2.msstate.edu/~fabel/pb52.pdf "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, [http://www2.msstate.edu/~fabel/tb37.pdf "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
* [http://eom.springer.de/B/b017470.htm Braid Theory] , "Encyclopaedia of Mathematics", Springer 2002
*Stephen Bigelow's [http://math.ucsb.edu/~bigelow/braids.html exploration of B5] Java applet.