Braid group
In mathematics, the braid group on n strands (denoted ), also known as the Artin braid group,[1] is the group whose elements are equivalence classes of nbraids (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]
Introduction
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 onetoone 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 . 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 spacetime trajectories formed by motion of physical rods, periodic orbits or "ghost rods", and almostinvariant sets has been used to estimate the topological entropy of several engineered and naturally occurring fluid systems, via the use of Nielsen–Thurston classification.[3][4][5]
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 of dimension at least 2. The symmetric product of copies of means the quotient of , the fold Cartesian product of by the permutation action of the symmetric group on strands operating on the indices of coordinates. That is, an ordered tuple is in the same orbit as any other that is a reordered version of it.
A path in the fold symmetric product is the abstract way of discussing points of , considered as an unordered tuple, independently tracing out strings. Since we must require that the strings never pass through each other, it is necessary that we pass to the subspace of the symmetric product, of orbits of tuples of distinct points. That is, we remove all the subspaces of defined by conditions for all . This is invariant under the symmetric group, and is the quotient by the symmetric group of the nonexcluded tuples. Under the dimension condition will be connected.
With this definition, then, we can call the braid group of with strings the fundamental group of (for any choice of base point – this is welldefined up to isomorphism). The case where is the Euclidean plane is the original one of Artin. In some cases it can be shown that the higher homotopy groups of 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.[6] A singlemove version of Markov's theorem, was published by in 1997.[7]
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.[8]
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.[9]
History
Braid groups were introduced explicitly by Emil Artin in 1925, although (as Wilhelm Magnus pointed out in 1974[10]) 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.[11] Braid groups are also understood by a deeper mathematical interpretation: as the fundamental group of certain configuration spaces.[11]
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.[12]
Basic properties
Generators and relations
Consider the following three braids:



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 's and their inverses.
It is clear that
 (i) ,
while the following two relations are not quite as obvious:
 (iia) ,
 (iib)
(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 and in the second group of relations, . 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.
Further properties
 The braid group is trivial, is an infinite cyclic group , and is isomorphic to the knot group of the trefoil knot – in particular, it is an infinite nonabelian group.
 The nstrand braid group embeds as a subgroup into the strand braid group by adding an extra strand that does not cross any of the first n strands. The increasing union of the braid groups with all is the infinite braid group .
 All nonidentity elements of have infinite order; i.e., is torsionfree.
 There is a leftinvariant linear order on called the Dehornoy order.
 For , contains a subgroup isomorphic to the free group on two generators.
 There is a homomorphism 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. Since σ_{i}^{k} ↦ k, then σ_{i}^{k} is the identity if and only if . This proves that the generators have infinite order.
Interactions
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 on n elements. This assignment is onto and compatible with composition, and therefore becomes a surjective group homomorphism B_{n} → S_{n} from the braid group onto the 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 of the symmetric group:
The kernel 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 same position. Pure braid groups fit into a short exact sequence
This sequence splits and therefore pure braid groups are realized as iterated semidirect products of free groups.
Relation between and the modular group
The braid group is the universal central extension of the modular group , 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 modulo its center, and equivalently, to the group of inner automorphisms of .
Here is a construction of this isomorphism. Define
 .
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 . Let denote the subgroup of 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
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 pseudoAnosov.
Connection to knot theory
If a braid is given and one connects the first lefthand item to the first righthand item using a new string, the second lefthand item to the second righthand 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
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_{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.
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 ntuples of objects or on the nfolded tensor product that involves some "twists". Consider an arbitrary group G and let X be the set of all ntuples 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_{i} is twisted by the inner automorphism 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 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 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 longstanding 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, Chetan 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 depending on the variables q and t. By suitably specializing these variables, the braid group 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 send the generators of to the first generators of (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 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
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:[13]
 .
So by definition
The calculations for coefficients in can be found in Fuks (1970).[14]
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 degreeone classes , subject to the relations[15]
See also
 Artin–Tits group
 Braided monoidal category
 Braided vector space
 Braided Hopf algebra
 Change ringing software – how software uses braid theory to model bellringing patterns
 Knot theory
 Noncommutative cryptography
References
 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 (2): 285–315. arXiv:alggeom/9608001. doi:10.1007/s000140050017.
 Boyland, Philip L.; Aref, Hassan; Stremler, Mark A. (2000), "Topological fluid mechanics of stirring" (PDF), Journal of Fluid Mechanics, 403 (1): 277–304, Bibcode:2000JFM...403..277B, doi:10.1017/S0022112099007107, MR 1742169, archived from the original (PDF) on 26 July 2011
 Gouillart, Emmanuelle; Thiffeault, JeanLuc; Finn, Matthew D. (2006), "Topological mixing with ghost rods", Physical Review E, 73 (3): 036311, arXiv:nlin/0510075, Bibcode:2006PhRvE..73c6311G, doi:10.1103/PhysRevE.73.036311, MR 2231368
 Stremler, Mark A.; Ross, Shane D.; Grover, Piyush; Kumar, Pankaj (2011), "Topological chaos and periodic braiding of almostcyclic sets" (PDF), Physical Review Letters, 106 (11): 114101, Bibcode:2011PhRvL.106k4101S, doi:10.1103/PhysRevLett.106.114101
 Markov, Andrey (1935), "Über die freie Äquivalenz der geschlossenen Zöpfe", Recueil Mathématique de la Société Mathématique de Moscou (in German and Russian), 1: 73–78
 Lambropoulou, Sofia; Rourke, Colin P. (1997), "Markov's theorem in 3manifolds", Topology and its Applications, 78 (1–2): 95–122, arXiv:math/0405498, doi:10.1016/S01668641(96)001514, MR 1465027
 Birman, Joan S. (1974), Braids, links, and mapping class groups, Annals of Mathematics Studies, 82, Princeton, N.J.: Princeton University Press, ISBN 9780691081496, MR 0375281
 Weisstein, Eric W. (August 2014). "Braid Index". MathWorld – A Wolfram Web Resource. Retrieved 6 August 2014.
 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 9783540068457.
 Artin, Emil (1947). "Theory of Braids". Annals of Mathematics. 48 (1): 101–126. doi:10.2307/1969218. JSTOR 1969218.
 Fox, Ralph; Neuwirth, Lee (1962). "The braid groups". Mathematica Scandinavica. 10: 119–126. doi:10.7146/math.scand.a10518. MR 0150755.
 Ghrist, Robert (1 December 2009). "Configuration Spaces, Braids, and Robotics". Braids. Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore. 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 (2): 143–151. doi:10.1007/BF01094491. MR 0274463.
 Arnol'd, Vladimir (1969). "The cohomology ring of the colored braid group" (PDF). Mat. Zametki. 5: 227–231. MR 0242196.
Further reading
 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
 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 00209910, MR 0422673
 Fox, Ralph; Neuwirth, Lee (1962), "The braid groups", Mathematica Scandinavica, 10: 119–126, doi:10.7146/math.scand.a10518, MR 0150755
 Kassel, Christian; Turaev, Vladimir (2008), Braid Groups, Springer, ISBN 9780387338415
 Menasco, William; Thistlethwaite, Morwen, eds. (2005), Handbook of Knot Theory, Elsevier, ISBN 9780444514523
External links
 "Braid group". PlanetMath.
 CRAG: CRyptography and Groups at Algebraic Cryptography Center Contains extensive library for computations with Braid Groups
 Visual Group Theory, Lecture 1.3: Groups in science, art, and mathematics
 Fabel, Paul (2005), "Completing Artin's braid group on infinitely many strands", Journal of Knot Theory and Its Ramifications, 14 (8): 979–991, arXiv:math/0201303, doi:10.1142/S0218216505004196, MR 2196643
 Fabel, Paul (2006), "The mapping class group of a disk with infinitely many holes", Journal of Knot Theory and Its Ramifications, 15 (1): 21–29, arXiv:math/0303042, doi:10.1142/S0218216506004324, MR 2204494
 Chernavskii, A.V. (2001) [1994], "Braid theory", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 9781556080104
 Bigelow, Stephen. "Exploration of B5 Java applet".
 Nayak, Chetan; Wilczek, Frank (1996), "$2n$ Quasihole States Realize $2^{n1}$Dimensional Spinor Braiding Statistics in Paired Quantum Hall States", Nuclear Physics B, 479 (3): 529–553, arXiv:condmat/9605145, Bibcode:1996NuPhB.479..529N, doi:10.1016/05503213(96)004300 — connection of projective braid group representations to the fractional quantum Hall effect
 Presentation for FradkinFest by Chetan V. Nayak
 Read, N. (2003), "Nonabelian braid statistics versus projective permutation statistics", Journal of Mathematical Physics, 44 (2): 558–563, arXiv:hepth/0201240, Bibcode:2003JMP....44..558R, doi:10.1063/1.1530369 — criticism of the reality of WilczekNayak 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.
 "Braids  the movie" A movie in computer graphics to explain some of braid theory (group presentation, word problem, closed braids and links, braids as motions of points in the plane).
 WINNER of Science magazine 2017 Dance Your PhD: Representations of the Braid Groups. Nancy Scherich.
 Behind the Math of Dance Your PhD, Part 1: The Braid Groups. Nancy Scherich. Explanation of braid groups as used in the movie.