Braid group

In mathematics, the braid group on strands (denoted formula_1), also known as the Artin braid group, is the group whose elements are equivalence classes of -braids (e.g. under ambient isotopy), and whose group operation is composition of braids (see ). 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 ); and in monodromy invariants of algebraic geometry.
Introduction.
In this introduction let ; the generalization to other values of 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:
On the other hand, two such connections which can be made to look the same by "pulling the strands" are considered "the same" braid:
All strands are required to move from left to right; knots like the following are "not" considered 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:
Another example:
The composition of the braids and is written as .
The set of all braids on four strands is denoted by formula_2. 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.)
Applications.
Braid theory has recently been applied to fluid mechanics, specifically to the field of chaotic mixing in fluid flows. The braiding of (2 + 1)-dimensional space-time trajectories formed by motion of physical rods, periodic orbits or "ghost rods", and almost-invariant sets has been used to estimate the topological entropy of several engineered and naturally occurring fluid systems, via the use of Nielsen–Thurston classification.
Another field of intense investigation involving braid groups and related topological concepts in the context of quantum physics is in the theory and (conjectured) experimental implementation of so-called anyons. These may well end up forming the basis for error-corrected quantum computing and so their abstract study is currently of fundamental importance in quantum information.
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. Alternatively, one can define the braid group purely algebraically via the braid relations, keeping the pictures in mind only to guide the intuition.
To explain how to reduce a braid group in the sense of Artin to a fundamental group, we consider a connected manifold formula_3 of dimension at least 2. The "symmetric product" of formula_4 copies of formula_3 means the quotient of formula_6, the formula_4-fold Cartesian product of formula_3 by the permutation action of the symmetric group on formula_4 strands operating on the indices of coordinates. That is, an ordered formula_4-tuple is in the same orbit as any other that is a re-ordered version of it.
A path in the formula_4-fold symmetric product is the abstract way of discussing formula_4 points of formula_3, considered as an unordered formula_4-tuple, independently tracing out formula_4 strings. Since we must require that the strings never pass through each other, it is necessary that we pass to the subspace formula_16 of the symmetric product, of orbits of formula_4-tuples of "distinct" points. That is, we remove all the subspaces of formula_6 defined by conditions formula_19 for all formula_20. This is invariant under the symmetric group, and formula_16 is the quotient by the symmetric group of the non-excluded formula_4-tuples. Under the dimension condition formula_16 will be connected.
With this definition, then, we can call the braid group of formula_3 with formula_4 strings the fundamental group of formula_16 (for any choice of base point – this is well-defined up to isomorphism). The case where formula_3 is the Euclidean plane is the original one of Artin. In some cases it can be shown that the higher homotopy groups of formula_16 are trivial.
Closed braids.
When "X" is the plane, the braid can be "closed", i.e., corresponding ends can be connected in pairs, to form a link, i.e., a possibly intertwined union of possibly knotted loops in three dimensions. The number of components of the link can be anything from 1 to "n", depending on the permutation of strands determined by the link. A theorem of J. W. Alexander demonstrates that every link can be obtained in this way as the "closure" of a braid. Compare with string links.
Different braids can give rise to the same link, just as different crossing diagrams can give rise to the same knot. In 1935, Andrey Markov Jr. described two moves on braid diagrams that yield equivalence in the corresponding closed braids. A single-move version of Markov's theorem, was published by in 1997.
Vaughan Jones originally defined his polynomial as a braid invariant and then showed that it depended only on the class of the closed braid.
The Markov theorem gives necessary and sufficient conditions under which the closures of two braids are equivalent links.
Braid index.
The "braid index" is the least number of strings needed to make a closed braid representation of a link. It is equal to the least number of Seifert circles in any projection of a knot.
History.
Braid groups were introduced explicitly by Emil Artin in 1925, although (as Wilhelm Magnus pointed out in 1974) they were already implicit in Adolf Hurwitz's work on monodromy from 1891.
Braid groups may be described by explicit presentations, as was shown by Emil Artin in 1947. Braid groups are also understood by a deeper mathematical interpretation: as the fundamental group of certain configuration spaces.
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.
Basic properties.
Generators and relations.
Consider the following three braids:
Every braid in formula_2 can be written as a composition of a number of these braids and their inverses. In other words, these three braids generate the group formula_2. To see this, an arbitrary braid is scanned from left to right for crossings; beginning at the top, whenever a crossing of strands formula_31 and formula_32 is encountered, formula_33 or formula_34 is written down, depending on whether strand formula_31 moves under or over strand formula_32. Upon reaching the right end, the braid has been written as a product of the formula_37's and their inverses.
It is clear that
while the following two relations are not quite as obvious:
(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 formula_41, formula_42 and formula_43 already follow from these relations and the group axioms.
Generalising this example to formula_4 strands, the group formula_1 can be abstractly defined via the following presentation:
where in the first group of relations formula_47 and in the second group of relations, formula_48. 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 equations.
Interactions.
Relation with symmetric group and the pure braid group.
By forgetting how the strands twist and cross, every braid on strands determines a permutation on elements. This assignment is onto and compatible with composition, and therefore becomes a surjective group homomorphism from the braid group onto the symmetric group. The image of the braid σ"i" ∈ is the transposition . 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:
The kernel of the homomorphism is the subgroup of called the pure braid group on strands and denoted . 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
This sequence splits and therefore pure braid groups are realized as iterated semi-direct products of free groups.
Relation between formula_52 and the modular group.
The braid group formula_52 is the universal central extension of the modular group formula_69, 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 formula_52 modulo its center, formula_72 and equivalently, to the group of inner automorphisms of formula_52.
Here is a construction of this isomorphism. Define
From the braid relations it follows that formula_75. Denoting this latter product as formula_76, one may verify from the braid relations that
implying that formula_76 is in the center of formula_52. Let formula_80 denote the subgroup of formula_52 generated by , since , it is a normal subgroup and one may take the quotient group . We claim ; this isomorphism can be given an explicit form. The cosets and map to
where and 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 to and to yields a surjective group homomorphism .
The center of is equal to , a consequence of the facts that is in the center, the modular group has trivial center, and the above surjective homomorphism has kernel .
Relationship to the mapping class group and classification of braids.
The braid group can be shown to be isomorphic to the mapping class group of a punctured disk with 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.
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 , this is often the preferred method of entering knots into computer programs.
Computational aspects.
The word problem for the braid relations is efficiently solvable and there exists a normal form for elements of in terms of the generators . (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 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.
In addition to the word problem, there are several known hard computational problems that could implement braid groups, applications in cryptography have been suggested.
Actions.
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 -tuples of objects or on the -folded tensor product that involves some "twists". Consider an arbitrary group and let be the set of all -tuples of elements of whose product is the identity element of . Then acts on in the following fashion:
Thus the elements and exchange places and, in addition, is twisted by the inner automorphism corresponding to — this ensures that the product of the components of remains the identity element. It may be checked that the braid group relations are satisfied and this formula indeed defines a group action of on . 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.
Elements of the braid group 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 . 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, Chetan Nayak and Frank Wilczek posited that in analogy to projective representations of , 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 formula_86 depending on the variables and . By suitably specializing these variables, the braid group formula_1 may be realized as a subgroup of the general linear group 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 to take the direct limit of braid groups, where the attaching maps formula_88 send the formula_89 generators of formula_90 to the first formula_89 generators of formula_55 (i.e., by attaching a trivial strand). Paul 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 formula_93 and the set of all braids—where a braid is defined to be a collection of paths from the points formula_94 to the points formula_95 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 formula_96 and to the fundamental group of the Hilbert cube minus the set
Cohomology.
The cohomology of a group formula_98 is defined as the cohomology of the corresponding Eilenberg–MacLane classifying space, formula_99, which is a CW complex uniquely determined by formula_98 up to homotopy. A classifying space for the braid group formula_1 is the th unordered configuration space of formula_102, that is, the set of formula_4 distinct unordered points in the plane: 
So by definition
The calculations for coefficients in formula_106 can be found in Fuks (1970).
Similarly, a classifying space for the pure braid group formula_96 is formula_108, the th "ordered" configuration space of formula_102. In 1968 Vladimir Arnold showed that the integral cohomology of the pure braid group formula_96 is the quotient of the exterior algebra generated by the collection of degree-one classes formula_111, subject to the relations