Editing Braid group

{{Short description|Group whose operation is a composition of braids}}
{{Use dmy dates|date=December 2023}}
[[File:5 Strand Braiding Technique.png|right|320px|thumbnail|A regular braid on five strands. Each arrow composes two further elements of <math>B_5</math>.]]
In [[mathematics]], the '''braid group on {{math|''n''}} strands''' (denoted <math>B_n</math>), also known as the '''Artin braid group''',<ref>{{Cite web|url=http://mathworld.wolfram.com/BraidGroup.html|title=Braid Group|last=Weisstein|first=Eric|website=Wolfram Mathworld}}</ref> is the group whose elements are equivalence classes of [[Braid theory|{{mvar|n}}-braids]] (e.g. under [[ambient isotopy]]), and whose [[group operation]] is composition of braids (see {{Section link||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 [[Emil Artin|Artin]]'s canonical presentation of the braid group corresponds to the [[Yang–Baxter equation]] (see {{Section link||Basic properties}}); and in [[monodromy]] invariants of [[algebraic geometry]].<ref>{{Cite journal|last1=Cohen|first1=Daniel|last2=Suciu|first2=Alexander|date=1997|title=The Braid Monodromy of Plane Algebraic Curves and Hyperplane Arrangements|journal=[[Commentarii Mathematici Helvetici]]|volume=72|issue=2|doi=10.1007/s000140050017|pages=285–315|arxiv=alg-geom/9608001|s2cid=14502859 }}</ref>

==Introduction==
In this introduction let {{math|''n'' {{=}} 4}}; the generalization to other values of {{math|''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:
:
{| valign="centre"
|-----
| [[File:braid s1 inv.png|The braid sigma 1<sup>−1</sup>]]
| &nbsp;&nbsp;&nbsp;is different from&nbsp;&nbsp;&nbsp;
<td>[[File:braid s1.png|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:
:
{| valign="centre"
|-----
| [[File:braid s1 inv.png|The braid sigma 1<sup>−1</sup>]]
| &nbsp;&nbsp;&nbsp; is the same as&nbsp;&nbsp;&nbsp;
<td>[[File:braid s1 inv alt.png|Another representation of sigma 1<sup>−1</sup>]]
|}

All strands are required to move from left to right; knots like the following are ''not'' considered braids:
:
{| valign="centre"
|-----
| [[File:braid nobraid.png|Not a braid]]
<td>&nbsp;&nbsp;&nbsp;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:
:
{| valign="centre"
|-----
| [[File:braid s3.png]]
| &nbsp;&nbsp;&nbsp; composed with &nbsp;&nbsp;&nbsp;
| [[File:braid s2.png]]
| &nbsp;&nbsp;&nbsp; yields &nbsp;&nbsp;&nbsp;
<td>[[File:braid s3s2.png]]
|}

Another example:
:
{| valign="centre"
|-----
| [[File:braid s1 inv s3 inv.png]]
| &nbsp;&nbsp;&nbsp; composed with &nbsp;&nbsp;&nbsp;
| [[File:braid s1 s3 inv.png]]
| &nbsp;&nbsp;&nbsp; yields &nbsp;&nbsp;&nbsp;
| [[File:braid s3 inv squared.png]]
|}
The composition of the braids {{math|σ}} and {{math|τ}} is written as {{math|στ}}.

The set of all braids on four strands is denoted by <math>B_4</math>. The above composition of braids is indeed a [[group (mathematics)|group]] operation. The [[identity element]] is the braid consisting of four parallel horizontal strands, and the [[inverse element|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&nbsp;+&nbsp;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]].<ref>{{citation|last1= Boyland|first1 = Philip L.|last2= Aref|first2 = Hassan|last3= Stremler|first3= Mark A.|bibcode = 2000JFM...403..277B|doi= 10.1017/S0022112099007107|mr= 1742169|journal     = Journal of Fluid Mechanics|pages = 277–304|title = Topological fluid mechanics of stirring|url= http://www.math.ufl.edu/~boyland/stir.pdf|volume = 403|issue = 1|year = 2000| hdl=2142/112556 | s2cid=47710742 |url-status = dead|archive-url= https://web.archive.org/web/20110726092237/http://www.math.ufl.edu/~boyland/stir.pdf|archive-date = 2011-07-26}}</ref><ref>{{citation| last1 = Gouillart | first1 = Emmanuelle| last2 = Thiffeault | first2 = Jean-Luc| last3 = Finn | first3 = Matthew D.| arxiv = nlin/0510075| doi = 10.1103/PhysRevE.73.036311| mr = 2231368| issue = 3| journal = Physical Review E| page = 036311| title = Topological mixing with ghost rods| volume = 73| year = 2006| pmid = 16605655|bibcode = 2006PhRvE..73c6311G | s2cid = 7142834}}</ref><ref>{{citation| last1 = Stremler | first1 = Mark A.| last2 = Ross | first2 = Shane D.| last3 = Grover | first3 = Piyush| last4 = Kumar | first4 = Pankaj| doi = 10.1103/PhysRevLett.106.114101| issue = 11| journal = Physical Review Letters| page = 114101| title = Topological chaos and periodic braiding of almost-cyclic sets| volume = 106| year = 2011| pmid = 21469863|bibcode = 2011PhRvL.106k4101S | hdl = 10919/24513| doi-access = free| hdl-access = free}}</ref>

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 the proposed particles [[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==
{{Main|Configuration space (mathematics)#Connection to braid groups}}

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 group]]s of a [[Configuration space (mathematics)|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]] <math>X</math> of dimension at least 2. The ''[[Symmetric product (topology)|symmetric product]]'' of <math>n</math> copies of <math>X</math> means the quotient of <math>X^n</math>, the <math>n</math>-fold [[Cartesian product]] of <math>X</math> by the permutation action of the [[symmetric group]] on <math>n</math> strands operating on the indices of coordinates. That is, an ordered <math>n</math>-tuple is in the same [[orbit (group theory)|orbit]] as any other that is a re-ordered version of it.

A path in the <math>n</math>-fold symmetric product is the abstract way of discussing <math>n</math> points of <math>X</math>, considered as an unordered <math>n</math>-tuple, independently tracing out <math>n</math> strings. Since we must require that the strings never pass through each other, it is necessary that we pass to the subspace <math>Y</math> of the symmetric product, of orbits of <math>n</math>-tuples of ''distinct'' points. That is, we remove all the subspaces of <math>X^n</math> defined by conditions <math>x_i = x_j</math> for all <math>1\le i<j\le n</math>. This is invariant under the symmetric group, and <math>Y</math> is the quotient by the symmetric group of the non-excluded <math>n</math>-tuples. Under the dimension condition <math>Y</math> will be connected.

With this definition, then, we can call '''the braid group of <math>X</math> with <math>n</math> strings''' the fundamental group of <math>Y</math> (for any choice of base point &ndash; this is well-defined [[up to]] isomorphism). The case where <math>X</math> is the Euclidean plane is the original one of Artin. In some cases it can be shown that the higher [[homotopy group]]s of <math>Y</math> are trivial.

===Closed braids=== <!-- [[closed braid]] and [[closed braids]] redirect to this section. -->
{{See also|Brunnian braid}}
When ''X'' is the plane, the braid can be ''closed'', i.e., corresponding ends can be connected in pairs, to form a [[link (knot theory)|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 [[Alexander's theorem|theorem]] of [[James Waddell Alexander II|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 (mathematics)|knot]]. In 1935, [[Andrey Markov Jr.]] described two moves on braid diagrams that yield equivalence in the corresponding closed braids.<ref>{{citation| last = Markov | first = Andrey | author-link = Andrey Markov Jr.| journal = Recueil Mathématique de la Société Mathématique de Moscou| language = de, ru| pages = 73–78| title = Über die freie Äquivalenz der geschlossenen Zöpfe| volume = 1| url = http://mi.mathnet.ru/msb5359| year = 1935}}</ref> A single-move version of Markov's theorem, was published by in 1997.<ref>{{citation| last1 = Lambropoulou | first1 = Sofia| last2 = Rourke | first2 = Colin P. | author-link2=Colin P. Rourke| doi = 10.1016/S0166-8641(96)00151-4| mr = 1465027| issue = 1–2| journal = [[Topology and Its Applications]]| pages = 95–122| title = Markov's theorem in 3-manifolds| volume = 78| year = 1997| arxiv = math/0405498| s2cid = 14494095}}</ref>

[[Vaughan Jones]] originally defined his [[Jones polynomial|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.<ref>{{citation| last = Birman | first = Joan S. | author-link = Joan Birman| isbn = 978-0-691-08149-6| mr = 0375281| location = Princeton, N.J.| publisher = [[Princeton University Press]]| series = Annals of Mathematics Studies| title = Braids, links, and mapping class groups|title-link= Braids, Links, and Mapping Class Groups | volume = 82| year = 1974}}</ref>

=== 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 circle]]s in any projection of a knot.<ref name="Wolfram">{{cite web |title=Braid Index |url=http://mathworld.wolfram.com/BraidIndex.html |author=Weisstein, Eric W. |author-link=Eric W. Weisstein |publisher=MathWorld – A Wolfram Web Resource |date=August 2014 |access-date=2014-08-06}}</ref>

==History==
Braid groups were introduced explicitly by [[Emil Artin]] in 1925, although (as [[Wilhelm Magnus]] pointed out in 1974<ref>{{cite book |series=Lecture Notes in Mathematics |volume=372 |first=Wilhelm |last=Magnus|author-link=Wilhelm Magnus |chapter=Braid groups: A survey |chapter-url=https://doi.org/10.1007%2FBFb0065203 |title=Proceedings of the Second International Conference on the Theory of Groups |publisher=Springer |year=1974 |isbn=978-3-540-06845-7 |pages=463–487 |doi=10.1007/BFb0065203 }}</ref>) they were already implicit in [[Adolf Hurwitz]]'s work on [[monodromy]] from 1891.

Braid groups may be described by explicit [[presentation of a group|presentation]]s, as was shown by [[Emil Artin]] in 1947.<ref name=Artin47>{{Cite journal|last=Artin|first=Emil|author-link=Emil Artin|date=1947|title=Theory of Braids|jstor=1969218|journal=[[Annals of Mathematics]]|volume=48| issue=1|pages=101–126|doi=10.2307/1969218}}</ref> Braid groups are also understood by a deeper mathematical interpretation: as the [[fundamental group]] of certain [[Configuration space (mathematics)|configuration space]]s.<ref name=Artin47/>

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.<ref>{{cite journal|last1=Fox|first1=Ralph|author1-link=Ralph Fox|last2= Neuwirth|first2= Lee|title=The braid groups|journal=Mathematica Scandinavica|volume= 10|year= 1962|pages= 119–126|mr=0150755|doi=10.7146/math.scand.a-10518|doi-access=free}}</ref>

==Basic properties==

===Generators and relations===
Consider the following three braids:

{|
|-----
| &nbsp;&nbsp;&nbsp;[[File:braid s1r.png]]&nbsp;&nbsp;&nbsp;
| &nbsp;&nbsp;&nbsp;[[File:braid s2r.png]]&nbsp;&nbsp;&nbsp;
|&nbsp;&nbsp;&nbsp;[[File:braid s3r.png]]&nbsp;&nbsp;&nbsp;
|-----
|<div class="center"><math>\sigma_1</math></div>
|<div class="center"><math>\sigma_2</math></div>
|<div class="center"><math>\sigma_3</math></div>
|}

Every braid in <math>B_4</math> can be written as a composition of a number of these braids and their inverses. In other words, these three braids [[generating set of a group|generate]] the group <math>B_4</math>.  To see this, an arbitrary braid is scanned from left to right for crossings. Numbering the strands beginning at the top, whenever a crossing of strands <math>i</math> and <math>i+1</math> is encountered, <math>\sigma_i</math> or <math>\sigma_i^{-1}</math> is written down, depending on whether strand <math>i</math> moves over or under strand <math>i+1</math>. Upon reaching the right end, the braid has been written as a product of the <math>\sigma_i</math> and their inverses.

It is clear that

::(i) <math>\sigma_1 \sigma_3 = \sigma_3 \sigma_1</math>,

while the following two relations are not quite as obvious:

::(iia) <math>\sigma_1 \sigma_2 \sigma_1 = \sigma_2 \sigma_1 \sigma_2</math>,
::(iib) <math>\sigma_2 \sigma_3 \sigma_2 = \sigma_3 \sigma_2 \sigma_3</math>

(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 <math>\sigma_1</math>, <math>\sigma_2</math> and <math>\sigma_3</math> already follow from these relations and the group axioms.

Generalising this example to <math>n</math> strands, the group <math>B_n</math> can be abstractly defined via the following [[presentation of a group|presentation]]:

:<math>B_n= \left \langle \sigma_1,\ldots,\sigma_{n-1}\mid \sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}, \sigma_i \sigma_j=\sigma_j\sigma_i \right \rangle, </math>

where in the first group of relations <math>1 \le i\le n-2</math> and in the second group of relations <math>|i-j|\ge 2</math>.<ref>{{Cite arXiv|title=BRAIDS: A SURVEY|first1=Joan|last1=Birman|first2=tara|last2=Brendle|date=2004 |page=1.2|eprint=math/0409205 }}</ref><ref>{{Cite web|url=https://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Lieber.pdf|title=Introduction to Braid Groups|last=Lieber|first=Joshua|page=4.1|website=math.uchicago.edu}}</ref> This presentation leads to generalisations of braid groups called [[Artin group]]s. The cubic relations, known as the '''braid relations''', play an important role in the theory of [[Yang–Baxter equation]]s.

===Further properties===
* The braid group <math>B_1</math> is [[trivial group|trivial]], <math>B_2</math> is the infinite [[cyclic group]] <math>\Z</math>, and <math>B_3</math> is isomorphic to the [[knot group]] of the [[trefoil knot]] – in particular, it is an infinite [[abelian group|non-abelian group]].
* The {{math|''n''}}-strand braid group <math>B_n</math> embeds as a [[subgroup]] into the <math>(n+1)</math>-strand braid group <math>B_{n+1}</math> by adding an extra strand that does not cross any of the first {{math|''n''}} strands. The increasing union of the braid groups with all <math>n \ge 1</math> is the '''infinite braid group''' <math>B_{\infty}</math>.
* All non-identity elements of <math>B_n</math> have infinite [[order (group theory)|order]]; i.e., <math>B_n</math> is [[torsion (algebra)|torsion-free]].
*There is a left-invariant [[linear order]] on <math>B_n</math> called the [[Dehornoy order]].
* For <math>n\ge 3</math>, <math>B_n</math> contains a subgroup isomorphic to the [[free group]] on two generators.
* There is a [[group homomorphism|homomorphism]] <math>B_n \to \Z</math> defined by {{math|σ<sub>''i''</sub> ↦ 1}}. So for instance, the braid {{math|σ<sub>2</sub>σ<sub>3</sub>σ<sub>1</sub><sup>−1</sup>σ<sub>2</sub>σ<sub>3</sub>}} is mapped to {{math|1&nbsp;+&nbsp;1&nbsp;−&nbsp;1&nbsp;+&nbsp;1&nbsp;+&nbsp;1 {{=}} 3}}. This map corresponds to the [[abelianization]] of the braid group. Since {{math|σ<sub>''i''</sub><sup>k</sup> ↦ k}}, then  {{math|σ<sub>''i''</sub><sup>k</sup>}} is the identity if and only if  <math>k=0</math>. 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 {{math|''n''}} strands determines a [[permutation]] on {{math|''n''}} elements. This assignment is onto and compatible with composition, and therefore becomes a [[surjective]] [[group homomorphism]] {{math|''B<sub>n</sub>'' → ''S<sub>n</sub>''}} from the braid group onto the [[symmetric group]]. The image of the braid σ<sub>''i''</sub> ∈ {{math|''B<sub>n</sub>''}} is the transposition {{math|''s''<sub>''i''</sub> {{=}} (''i'', ''i''+1) ∈ ''S<sub>n</sub>''}}. 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 group|Coxeter presentation]] of the symmetric group:

:<math>S_n = \left \langle s_1,\ldots,s_{n-1}| s_i s_{i+1} s_i=s_{i+1} s_i s_{i+1}, s_i s_j = s_j s_i \text{ for } |i-j|\geq 2, s_i^2=1 \right\rangle. </math>

The [[kernel (algebra)|kernel]] of the homomorphism {{math|''B<sub>n</sub>'' → ''S<sub>n</sub>''}} is the subgroup of {{math|''B<sub>n</sub>''}} called the '''pure braid group on {{math|''n''}} strands''' and denoted {{math|''P<sub>n</sub>''}}. This can be seen as the fundamental group of the space of {{math|''n''}}-tuples of distinct points of the Euclidean plane. 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]]

: <math>1\to F_{n-1} \to P_n \to P_{n-1}\to 1.</math>

This sequence splits and therefore pure braid groups are realized as iterated [[semidirect product|semi-direct products]] of free groups.

===Relation between B<sub>3</sub> and the modular group===
[[File:Braid-modular-group-cover.svg|thumb|376px|<math>B_3</math> is the [[universal central extension]] of the modular group.]]

The braid group <math>B_3</math> is the [[universal central extension]] of the [[modular group]] <math>\mathrm{PSL}(2, \Z)</math>, with these sitting as lattices inside the (topological) universal covering group

:<math>\overline{\mathrm{SL}(2,\R)} \to \mathrm{PSL}(2,\R)</math>.

Furthermore, the modular group has trivial center, and thus the modular group is isomorphic to the [[quotient group]] of <math>B_3</math> modulo its [[center (group theory)|center]], <math>Z(B_3),</math> and equivalently, to the group of [[inner automorphism]]s of <math>B_3</math>.

Here is a construction of this [[isomorphic|isomorphism]]. Define

:<math>a = \sigma_1 \sigma_2 \sigma_1, \quad b = \sigma_1 \sigma_2</math>.

From the braid relations it follows that <math>a^2 = b^3</math>. Denoting this latter product as <math>c</math>, one may verify from the braid relations that

:<math>\sigma_1 c \sigma_1^{-1} = \sigma_2 c \sigma_2^{-1}=c</math>

implying that <math>c</math> is in the center of <math>B_3</math>. Let <math>C</math> denote the [[subgroup]] of <math>B_3</math> [[group generator|generated]] by {{math|''c''}}, since {{math|''C''&nbsp;⊂&nbsp;''Z''(''B''<sub>3</sub>)}}, it is a [[normal subgroup]] and one may take the [[quotient group]] {{math|''B''<sub>3</sub>/''C''}}. We claim {{math|''B''<sub>3</sub>/''C'' ≅ PSL(2, '''Z''')}}; this isomorphism can be given an explicit form. The [[coset]]s {{math|σ<sub>1</sub>''C''}} and {{math|σ<sub>2</sub>''C''}} map to

:<math>\sigma_1C \mapsto R=\begin{bmatrix}1 & 1 \\ 0 & 1 \end{bmatrix} \qquad \sigma_2C \mapsto L^{-1}=\begin{bmatrix}1 & 0 \\ -1 & 1 \end{bmatrix}</math>

where {{math|''L''}} and {{math|''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 of a group|presentation]] for the modular group is

:<math>\langle v,p\, |\, v^2=p^3=1\rangle</math>

where

:<math>v=\begin{bmatrix}0 & 1 \\ -1 & 0 \end{bmatrix}, \qquad p=\begin{bmatrix}0 & 1 \\ -1 & 1 \end{bmatrix}.</math>

Mapping {{math|''a''}} to {{math|''v''}} and {{math|''b''}} to {{math|''p''}} yields a surjective group homomorphism {{math|''B''<sub>3</sub> → PSL(2, '''Z''')}}.

The center of {{math|''B''<sub>3</sub>}} is equal to {{math|''C''}}, a consequence of the facts that {{math|''c''}} is in the center, the modular group has trivial center, and the above surjective homomorphism has [[kernel (algebra)|kernel]] {{math|''C''}}.

===Relationship to the mapping class group and classification of braids===
The braid group {{math|''B<sub>n</sub>''}} can be shown to be isomorphic to the [[mapping class group]] of a [[punctured disk]] with {{math|''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 [[Nielsen–Thurston classification|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 (knot theory)|link]], and sometimes a [[knot (mathematics)|knot]]. [[Alexander's theorem]] in [[braid theory]] states that the converse is true as well: every [[knot (mathematics)|knot]] and every [[link (knot theory)|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 {{math|σ<sub>''i''</sub>}}, this is often the preferred method of entering knots into computer programs.

=== Computational aspects ===
The [[word problem for groups|word problem]] for the braid relations is efficiently solvable and there exists a [[Normal form (abstract rewriting)|normal form]] for elements of {{math|''B<sub>n</sub>''}} in terms of the generators {{math|σ<sub>1</sub>, ..., σ<sub>''n''−1</sub>}}. (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]] can carry out computations in {{math|''B<sub>n</sub>''}} 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.<ref>{{cite arXiv |last=Garber |first=David |eprint=0711.3941v2 |title=Braid Group Cryptography |class=cs.CR |date=2009 }}</ref>

==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 {{math|''n''}}-tuples of objects or on the {{math|''n''}}-folded [[tensor product]] that involves some "twists". Consider an arbitrary group {{math|''G''}} and let {{math|''X''}} be the set of all {{math|''n''}}-tuples of elements of {{math|''G''}} whose product is the [[identity element]] of {{math|''G''}}. Then {{math|''B<sub>n</sub>''}} [[Group action (mathematics)|acts]] on {{math|''X''}} in the following fashion:

: <math> \sigma_i \left (x_1,\ldots,x_{i-1},x_i, x_{i+1},\ldots, x_n \right)= \left (x_1,\ldots, x_{i-1}, x_{i+1}, x_{i+1}^{-1}x_i x_{i+1}, x_{i+2},\ldots,x_n \right ). </math>

Thus the elements {{math|''x<sub>i</sub>''}} and {{math|''x''<sub>''i''+1</sub>}} exchange places and, in addition, {{math|''x<sub>i</sub>''}} is twisted by the [[inner automorphism]] corresponding to {{math|''x''<sub>''i''+1</sub>}} – this ensures that the product of the components of {{math|''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 {{math|''B<sub>n</sub>''}} on {{math|''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 invariant]]s.

===Representations===
Elements of the braid group {{math|''B<sub>n</sub>''}} can be represented more concretely by matrices. One classical such [[group representation|representation]] is [[Burau representation]], where the matrix entries are single variable [[Laurent polynomial]]s. It had been a long-standing question whether Burau representation was [[faithful representation|faithful]], but the answer turned out to be negative for {{math|''n''&nbsp;≥&nbsp;5}}. More generally, it was a major open problem whether braid groups were [[linear group|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 {{math|SO(3)}}, the projective representations of the braid group have a physical meaning for certain quasiparticles in the [[fractional quantum hall effect]].<ref>{{citation |first1=Chetan |last1=Nayak |first2=Frank |last2=Wilczek | author-link2=Frank Wilczek|title={{math|2''n''}} Quasihole States Realize {{math|2<sup>''n''-1</sup>}}-Dimensional Spinor Braiding Statistics in Paired Quantum Hall States |date=1996 |doi=10.1016/0550-3213(96)00430-0 |arxiv=cond-mat/9605145 |volume=479 |issue=3 |journal=Nuclear Physics B |pages=529–553|bibcode=1996NuPhB.479..529N |s2cid=18726223 }}  Some of Wilczek-Nayak's proposals subtly violate known physics; see the discussion {{citation |first=N. |last=Read |title=Nonabelian braid statistics versus projective permutation statistics |journal=Journal of Mathematical Physics |volume=44 |issue=2 |pages=558–563 |year=2003 |arxiv=hep-th/0201240 |doi=10.1063/1.1530369|bibcode=2003JMP....44..558R |s2cid=119388336 }}</ref> Around 2001 [[Stephen Bigelow]] and Daan Krammer independently proved that all braid groups are linear. Their work used the [[Lawrence–Krammer representation]] of dimension <math>n(n-1)/2</math> depending on the variables {{math|''q''}} and {{math|''t''}}. By suitably specializing these variables, the braid group <math>B_n</math> 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 <math>f \colon B_n \to B_{n+1}</math> send the <math>n-1</math> generators of <math>B_{n}</math> to the first <math>n-1</math>  generators of <math>B_{n+1}</math> (i.e., by attaching a trivial strand). This group, however, admits no metrizable topology while remaining continuous.

Paul Fabel has shown that there are two [[topological space|topologies]] that can be imposed on the resulting group each of whose [[complete metric space|completion]] yields a different group.<ref>
*{{citation |first=Paul |last=Fabel |title=Completing Artin's braid group on infinitely many strands |journal=Journal of Knot Theory and Its Ramifications |volume=14 |issue=8 |pages=979–991 |year=2005 |doi=10.1142/S0218216505004196|arxiv=math/0201303 |mr=2196643|s2cid=16998867 }}
*{{citation |first=Paul |last=Fabel |title=The mapping class group of a disk with infinitely many holes |journal=Journal of Knot Theory and Its Ramifications |volume=15 |issue=1 |pages=21–29 |year=2006 |doi=10.1142/S0218216506004324 |arxiv=math/0303042|mr=2204494 |s2cid=13892069 }}</ref> The first 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 [[unit disk|disk]].

The second group can be thought of the same as with finite braid groups.  Place a strand at each of the points <math>(0, 1/n)</math> and the set of all braids—where a braid is defined to be a collection of paths from the points <math>(0, 1/n, 0)</math> to the points <math>(0, 1/n, 1)</math> 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 <math>P_n</math> and to the [[fundamental group]] of the [[Hilbert cube]] minus the set
:<math>\{(x_i)_{i\in \mathbb{N}} \mid x_i=x_j\text{ for some }i\ne j\}.</math>

==Cohomology==
{{See also|Configuration space (mathematics)#Connection to braid groups}}

The [[Group cohomology|cohomology of a group]] <math>G</math> is defined as the cohomology of the corresponding [[Eilenberg–MacLane space|Eilenberg–MacLane]] [[classifying space]], <math>K(G, 1)</math>, which is a [[CW complex]] uniquely determined by <math>G</math> up to homotopy. A classifying space for the braid group <math>B_n</math> is the {{var|n}}<sup>th</sup> unordered [[Configuration space (mathematics)|configuration space]] of <math>\R^2</math>, that is, the space of all sets of <math>n</math> distinct unordered points in the plane:<ref>{{Cite book|title=Braids|last=Ghrist|first=Robert|author-link=Robert Ghrist|date=2009-12-01|publisher=[[World Scientific]]|
isbn=9789814291408|series=Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore|volume=19|pages=263–304|doi=10.1142/9789814291415_0004|chapter = Configuration Spaces, Braids, and Robotics}}</ref> 
:<math>\operatorname{UConf}_n(\R^2) = \{ \{u_1, ..., u_n\} : u_i \in \R^2, u_i \neq u_j \text{ for } i \neq j \}</math>.
So by definition

The calculations for coefficients in <math>\Z/2\Z</math> can be found in Fuks (1970).<ref>{{cite journal|last=Fuks|first=Dmitry B.| author-link=Dmitry Fuchs|title=Cohomology of the braid group mod 2| journal=Functional Analysis and Its Applications |volume=4|issue=2|year=1970 |pages=143–151| doi=10.1007/BF01094491| mr=0274463|s2cid=123442457 |language=en}}</ref>

Similarly, a classifying space for the pure braid group <math>P_n</math> is <math>\operatorname{Conf}_n(\R^2)</math>, the {{var|n}}<sup>th</sup> ''ordered'' [[Configuration space (mathematics)|configuration space]] of <math>\R^2</math>. In 1968 [[Vladimir Arnold]] showed that the integral cohomology of the pure braid group <math>P_n</math> is the quotient of the [[exterior algebra]] generated by the collection of degree-one classes <math>\omega_{ij} \; \; 1 \leq i < j \leq n</math>, subject to the relations<ref>{{cite journal|last=Arnol'd|first=Vladimir|author-link=Vladimir Arnold|title=The cohomology ring of the colored braid group|journal=Mat. Zametki |volume=5|year= 1969|pages= 227–231|mr=0242196 |
url=http://www.pdmi.ras.ru/~arnsem/Arnold/arnold_MZ69e.pdf}}</ref>
:<math>\omega_{k,\ell} \omega_{\ell,m} + \omega_{\ell,m} \omega_{m,k} + \omega_{m,k} \omega_{k,\ell} =0.</math>

==See also==
*[[Artin–Tits group]]
*[[Braided monoidal category]]
*[[Braided vector space]]
*[[Braided Hopf algebra]]
*[[Knot theory]]
*[[Non-commutative cryptography]]
*[[Spherical braid group]]

==References==
<references/>

==Further reading==
{{refbegin}}
*{{citation |last1=Birman |first1=Joan |author-link=Joan Birman| last2=Brendle |first2=Tara E.|author2-link= Tara E. Brendle |title=Braids: A Survey |date=26 February 2005 |arxiv=math.GT/0409205}}. In {{harvnb|Menasco|Thistlethwaite|2005}}
*{{citation |last1=Carlucci |first1=Lorenzo |last2=Dehornoy |first2=Patrick |author-link2=Patrick Dehornoy|last3=Weiermann |first3=Andreas |title=Unprovability results involving braids |journal=[[Proceedings of the London Mathematical Society]] | 
series=3|volume= 102 |year=2011|issue= 1|pages= 159–192 |arxiv=0711.3785 |doi=10.1112/plms/pdq016|mr=2747726|s2cid=16467487 }}
*{{SpringerEOM| title=Braid theory | id=Braid_theory | oldid=1855 | first=A.V. | last=Chernavskii }}
*{{Citation | last1=Deligne | first1=Pierre | author1-link=Pierre Deligne | title=Les immeubles des groupes de tresses généralisés | doi=10.1007/BF01406236 | mr=0422673 | year=1972 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=17 | pages=273–302 | issue=4| bibcode=1972InMat..17..273D | s2cid=123680847 }}
*{{citation| last1 = Fox | first1 = Ralph | author1-link = Ralph Fox| last2 = Neuwirth | first2 = Lee| mr = 0150755| journal = Mathematica Scandinavica| pages = 119–126| title = The braid groups| volume = 10| year = 1962| doi = 10.7146/math.scand.a-10518| doi-access = free}}
*{{citation |last1=Kassel |first1=Christian |last2=Turaev |first2=Vladimir |author-link2=Vladimir Turaev|title=Braid Groups |publisher=Springer |year=2008 |isbn=978-0-387-33841-5 |url=https://books.google.com/books?id=y6Cox3XjdroC}}
*{{citation |editor1-last=Menasco |editor1-first=William |editor1-link= William Menasco | editor2-last=Thistlethwaite |editor2-first=Morwen |editor2-link= Morwen Thistlethwaite| title=Handbook of Knot Theory |publisher=Elsevier |year=2005 |isbn=978-0-444-51452-3 }}
{{refend}}

== External links ==
*{{planetmath reference|urlname=BraidGroup|title=Braid group}}
*[http://web.stevens.edu/algebraic/downloads.php CRAG: CRyptography and Groups] computation library from the [[Stevens University]]'s [http://www.acc.stevens.edu Algebraic Cryptography Center]
*{{cite serial |first=M. |last=Macauley |series=Visual Group Theory |title=Lecture 1.3: Groups in science, art, and mathematics |url=https://www.youtube.com/watch?v=6NElZENlMjI |publisher=Clemson University}}
*{{cite web|first=Stephen|last=Bigelow|author-link=Stephen Bigelow|url=http://math.ucsb.edu/~bigelow/braids.html|title=Exploration of B5 Java applet|access-date=1 November 2007|archive-date=4 June 2013|archive-url=https://web.archive.org/web/20130604224001/http://www.math.ucsb.edu/~bigelow/braids.html|url-status=dead}}
*{{citation |first=Helger |last=Lipmaa |title=Cryptography and Braid Groups page |url=http://research.cyber.ee/~lipmaa/crypto/link/public/braid/ |url-status=dead |archive-url=https://web.archive.org/web/20090803144521/http://research.cyber.ee/~lipmaa/crypto/link/public/braid/ |archive-date=3 August 2009 }}
*{{cite AV media |url=https://www.youtube.com/playlist?list=PLyxHTRWELFBoijzAvXTDOLIF5yfkwN4wZ |title=Braids – the movie |first=Ester |last=Dalvit |date=2015}}
*{{cite serial |first=Nancy |last=Scherich |url=https://www.youtube.com/watch?v=MASNukczu5A |series=Dance Your PhD |title=Representations of the Braid Groups}} expanded further in [https://www.youtube.com/watch?v=1za9h2S98Wk Behind the Math of "Dance Your PhD," Part 1: The Braid Groups.]

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{{Braiding}}
{{Knot theory|state=collapsed}}

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[[Category:Braid groups| ]]
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[[Category:Diagram algebras]]

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