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| | Adrianne is what you is able to call me but Which i don't like when guys and women use my full address. What I love doing would be to play [http://Browse.Deviantart.com/?qh=§ion=&global=1&q=croquet croquet] together with now I have your time to take on new things. The job I've been occupying of years is an choose clerk. My husband and I select to reside in Guam but I will have to move in a year or two. You can sometimes find my website here: http://circuspartypanama.com<br><br>my weblog; [http://circuspartypanama.com clash of clans hack for mac] |
| In [[mathematics]], the '''braid group on ''n'' strands''', denoted by ''B''<sub>''n''</sub>, is a [[group (mathematics)|group]] which has an intuitive geometrical representation, and in a sense generalizes the [[symmetric group]] ''S''<sub>''n''</sub>. Here, ''n'' is a [[natural number]]; if ''n'' > 1, then ''B''<sub>''n''</sub> 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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| ==Introduction==
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| ===Intuitive description===
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| 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:
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| :
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| {| valign="centre"
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| |-----
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| | [[File:braid s1 inv.png|The braid sigma_1^(-1)]]
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| | is different from
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| <td>[[File:braid s1.png|The braid sigma_1]]
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| |}
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| On the other hand, two such connections which can be made to look the same by "pulling the strands" are considered ''the same'' braid:
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| :
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| {| valign="centre"
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| |-----
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| | [[File:braid s1 inv.png|The braid sigma_1^(-1)]]
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| | is the same as
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| <td>[[File:braid s1 inv alt.png|Another representation of sigma_1^(-1)]]
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| |}
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| All strands are required to move from left to right; knots like the following are ''not'' considered braids:
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| :
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| {| valign="centre"
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| |-----
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| | [[File:braid nobraid.png|Not a braid]]
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| <td> is not a braid
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| |}
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| 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:
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| :
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| {| valign="centre"
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| |-----
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| | [[File:braid s3.png]]
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| | composed with
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| | [[File:braid s2.png]]
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| | yields
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| <td>[[File:braid s3s2.png]]
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| |}
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| Another example:
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| :
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| {| valign="centre"
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| |-----
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| | [[File:braid s1 inv s3 inv.png]]
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| | composed with
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| | [[File:braid s1 s3 inv.png]]
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| | yields
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| | [[File:braid s3 inv squared.png]]
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| |}
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| The composition of the braids σ and τ is written as στ.
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| The set of all braids on four strands is denoted by ''B''<sub>4</sub>. 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.)
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| ===Formal treatment===
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| 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]]. This is outlined in the article on [[braid theory]].
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| Alternatively, one can define the braid group purely algebraically via the braid relations, keeping the pictures in mind only to guide the intuition.
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| ===History===
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| Braid groups were introduced explicitly by [[Emil Artin]] in 1925, although (as [[Wilhelm Magnus]] pointed out in 1974<ref>Wilhelm Magnus. [http://www.springerlink.com/content/2w7lp1402314k343/?p=8ca127a073654ea182c15cbecdeca393&pi=3 Braid groups: A survey]. In ''Lecture Notes in Mathematics'', volume 372, pages 463–487. Springer, 1974. ''Proceedings of the Second International Conference on the Theory of Groups'', Canberra, Australia, 1973. ISBN 978-3-540-06845-7</ref>) 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.
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| ==Basic properties==
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| ===Generators and relations===
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| Consider the following three braids:
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| {|
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| |-----
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| | [[File:braid s1.png]]
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| | [[File:braid s2.png]]
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| <td> [[File:braid s3.png]]
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| |-----
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| <td><center>σ<sub>1</sub></center>
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| <td><center>σ<sub>2</sub></center>
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| <td><center>σ<sub>3</sub></center>
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| |}
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| Every braid in ''B''<sub>4</sub> 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 ''B''<sub>4</sub>. To see this, an arbitrary braid is scanned from left to right for crossings; beginning at the top, whenever a crossing of strands ''i'' and ''i'' + 1 is encountered, σ<sub>''i''</sub> or σ<sub>''i''</sub><sup>−1</sup> 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.
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| It is clear that
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| (i):σ<sub>1</sub>σ<sub>3</sub> = σ<sub>3</sub>σ<sub>1</sub>,
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| while the following two relations are not quite as obvious:
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| (iia):σ<sub>1</sub>σ<sub>2</sub>σ<sub>1</sub> = σ<sub>2</sub>σ<sub>1</sub>σ<sub>2</sub>,
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| (iib):σ<sub>2</sub>σ<sub>3</sub>σ<sub>2</sub> = σ<sub>3</sub>σ<sub>2</sub>σ<sub>3</sub>
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| (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 σ<sub>1</sub>, σ<sub>2</sub> and σ<sub>3</sub> already follow from these relations and the group axioms.
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| Generalising this example to ''n'' strands, the group ''B''<sub>''n''</sub> can be abstractly defined via the following [[presentation of a group|presentation]]:
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| : <math> B_n=\langle \sigma_1,\ldots,\sigma_{n-1}|
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| \sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1},
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| \sigma_i\sigma_j=\sigma_j\sigma_i \rangle, </math>
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| where in the first group of relations 1 ≤ ''i'' ≤ ''n''−2 and in the second group of relations, |''i'' − ''j''| ≥ 2. 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]].
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| ===Further properties===
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| * The braid group ''B''<sub>1</sub> is [[trivial group|trivial]], ''B''<sub>2</sub> is an infinite [[cyclic group]] '''Z''', and ''B''<sub>3</sub> is isomorphic to the [[knot group]] of the [[trefoil knot]] – in particular, it is an infinite [[abelian group|non-abelian group]].
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| * The ''n''-strand braid group ''B''<sub>''n''</sub> embeds as a [[subgroup]] into the (''n''+1)-strand braid group ''B''<sub>''n''+1</sub> by adding an extra strand that does not cross any of the first ''n'' strands. The increasing union of the braid groups with all ''n'' ≥ 1 is the '''infinite braid group''' ''B''<sub>∞</sub>.
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| * All non-identity elements of ''B''<sub>''n''</sub> have infinite [[order (group theory)|order]]; i.e., ''B''<sub>''n''</sub> is [[torsion (algebra)|torsion-free]].
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| *There is a left-invariant [[linear order]] on ''B''<sub>''n''</sub> called the [[Dehornoy order]].
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| * For ''n'' ≥ 3, ''B''<sub>''n''</sub> contains a subgroup isomorphic to the [[free group]] on two generators.
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| * There is a [[group homomorphism|homomorphism]] ''B''<sub>''n''</sub> → '''Z''' that maps every σ<sub>''i''</sub> to 1. So for instance, the braid 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 1 + 1 − 1 + 1 + 1 = 3.
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| ==Interactions of braid groups==
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| ===Relation with symmetric group and the pure braid group===
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| By forgetting how the strands twist and cross, every braid on ''n'' strands determines a [[permutation]] on ''n'' elements. This assignment is onto, compatible with composition, and therefore becomes a [[surjective]] [[group homomorphism]] ''B''<sub>''n''</sub> → ''S''<sub>''n''</sub> from the braid group into the [[symmetric group]]. The image of the braid σ<sub>''i''</sub> ∈ ''B''<sub>''n''</sub> is the transposition ''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:
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| : <math> S_n=\langle s_1,\ldots,s_{n-1}|
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| s_i s_{i+1} s_i=s_{i+1} s_i s_{i+1},
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| s_i s_j = s_j s_i ~\rm{for}~|i-j|\geq 2, s_i^2=1 \rangle. </math>
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| The [[kernel (algebra)|kernel]] of the homomorphism ''B''<sub>''n''</sub> → ''S''<sub>''n''</sub> is the subgroup of ''B''<sub>''n''</sub> called the '''pure braid group on ''n'' strands''' and denoted ''P''<sub>''n''</sub>. In a pure braid, the beginning and the end of each strand are in the like positions. Pure braid groups fit into a [[short exact sequence]]
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| : <math>1\to F_{n-1} \to P_n \to P_{n-1}\to 1.</math>
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| This sequence splits and therefore pure braid groups are realized as iterated [[semidirect product|semi-direct products]] of free groups.
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| ===Relation between ''B''<sub>3</sub> and the modular group===
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| [[File:Braid-modular-group-cover.svg|thumb|376px|''B''<sub>3</sub> is the [[universal central extension]] of the modular group.]]
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| The braid group ''B''<sub>3</sub> is the [[universal central extension]] of the [[modular group]] PSL(2,'''Z'''), with these sitting as lattices inside the (topological) universal covering group <math>\overline{\mathrm{SL}(2,\mathbf{R})} \to \mathrm{PSL}(2,\mathbf{R}).</math>
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| Further, 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]];
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| equivalently, to the group of [[inner automorphism]]s of <math>B_3</math>.
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| A construction is given below.
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| Define <math>a = \sigma_1 \sigma_2 \sigma_1</math> and
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| <math>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=a^2=b^3</math>, one may verify from the braid relations that
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| :<math>\sigma_1 c \sigma_1^{-1} = \sigma_2 c \sigma_2^{-1}=c</math>
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| implying that <math>c</math> is in the center of ''B''<sub>3</sub>. The [[subgroup]] <math>\langle c\rangle</math> of ''B''<sub>3</sub> [[group generator|generated]] by <math>c</math> is therefore a [[normal subgroup]]. Since it is normal, one may take the [[quotient group]]; this quotient group is [[isomorphic]] to the modular group:
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| :<math>PSL(2,\mathbf{Z}) \simeq B_3/\langle c\rangle.</math>
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| This isomorphism can be given an explicit form. The [[coset]]s <math>[\sigma_1]</math> of <math>\sigma_1</math> and <math>[\sigma_2]</math> of <math>\sigma_2</math> map to
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| :<math>[\sigma_1] \mapsto R=\begin{bmatrix}1 & 1 \\ 0 & 1 \end{bmatrix}
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| \qquad [\sigma_2] \mapsto L^{-1}=\begin{bmatrix}1 & 0 \\ -1 & 1 \end{bmatrix}
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| </math>
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| where <math>L</math> and <math>R</math> are the standard left and right moves on the [[Stern-Brocot tree]]; it is well known that these moves generate the modular group.
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| Alternately, one common [[presentation of a group|presentation]] for the modular group is
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| :<math>\langle v,p\, |\, v^2=p^3=1\rangle</math>
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| where
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| :<math>
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| v=\begin{bmatrix}0 & 1 \\ -1 & 0 \end{bmatrix}, \qquad p=\begin{bmatrix}0 & 1 \\ -1 & 1 \end{bmatrix}.
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| </math>
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| Mapping ''a'' to ''v'' and ''b'' to ''p'' yields a surjective group homomorphism from ''B''<sub>3</sub> to PSL(2,'''Z''').
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| The center of ''B''<sub>3</sub> is equal to <math>\langle c\rangle</math>, a consequence of the facts that ''c'' is in the center, the modular group has trivial center, and the above surjective homomorphism has [[kernel (algebra)|kernel]] <math>\langle c\rangle</math>.
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| ===Relationship to the mapping class group and classification of braids===
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| The braid group ''B''<sub>n</sub> 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 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.
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| Via this mapping class group interpretation of braids, each braid may be classified as [[Nielsen-Thurston classification|periodic, reducible or pseudo-Anosov]].
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| ===Connection to knot theory===
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| 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 σ<sub>''i''</sub>, this is often the preferred method of entering knots into computer programs.
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| ===Computational aspects===
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| 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 ''B''<sub>''n''</sub> in terms of the generators 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]] can carry out computations in ''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.
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| Since there are nevertheless several hard computational problems about braid groups, applications in [[cryptography]] have been suggested.
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| ==Actions of braid groups==
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| In analogy with the action of the symmetric group by permutations, in various mathematical settings there exists a natural action the braid group on ''n''-tuples of objects or on the ''n''-folded [[tensor product]] that involves some "twists". Consider an arbitrary group ''G'' and let ''X'' be the set of all ''n''-tuples of elements of ''G'' whose product is the [[identity element]] of ''G''. Then ''B''<sub>''n''</sub> [[group action|acts]] on ''X'' in the following fashion:
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| : <math> \sigma_i(x_1,\ldots,x_{i-1},x_i, x_{i+1},\ldots, x_n)=
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| (x_1,\ldots, x_{i-1}, x_{i+1}, x_{i+1}^{-1}x_i x_{i+1}, x_{i+2},\ldots,x_n).
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| </math>
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| Thus the elements ''x''<sub>''i''</sub> and ''x''<sub>''i''+1</sub> exchange places and, in addition, ''x''<sub>''i''</sub> is twisted by the [[inner automorphism]] corresponding to ''x''<sub>''i''+1</sub> — 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''<sub>''n''</sub> 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 invariant]]s.
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| ===Representations===
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| Elements of the braid group ''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 ''n'' ≥ 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, C. 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 ''n''(''n''−1)/2 depending on the variables ''q'' and ''t''. By suitably specializing these variables, the braid group ''B''<sub>''n''</sub> may be realized as a subgroup of the [[general linear group]] over the [[complex numbers]].
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| ==Infinitely generated braid groups==
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| 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 <math>f: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). 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. 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 [[unit disk|disk]].
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| 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 \Bbb{N}} \mid x_i=x_j\text{ for some }i\ne j\}</math>.
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| ==Notes==
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| <references/>
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| ===References===
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| *{{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}}
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| ===Further reading===
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| * Birman, Joan, and Brendle, Tara E., [http://arxiv.org/abs/math.GT/0409205 "Braids: A Survey"], revised 26 February 2005. In Menasco and Thistlethwaite.
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| * Carlucci, Lorenzo; Dehornoy, Patrick; and Weiermann, Andreas, [http://arxiv.org/abs/0711.3785 "Unprovability results involving braids"], 23 November 2007
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| * Kassel, Christian; and Turaev, Vladimir, [http://books.google.com/books?id=y6Cox3XjdroC&dq=turaev+braid+groups&printsec=frontcover&source=bl&ots=eA5ZkHtkhx&sig=s6mXDyKHcpJVRFiqb1B5VRbA_Yc&hl=en&sa=X&oi=book_result&resnum=1&ct=result ''Braid Groups''], Springer, 2008. ISBN 0-387-33841-1
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| * Menasco, W., and Thistlethwaite, M., (editors), ''Handbook of Knot Theory'', Amsterdam : Elsevier, 2005. ISBN 0-444-51452-X
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| == External links ==
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| *{{planetmath reference|id=4604|title=Braid group}}
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| *[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
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| *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
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| *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
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| *{{SpringerEOM| title=Braid theory | id=Braid_theory | oldid=1855 | first=A.V. | last=Chernavskii }}
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| *Stephen Bigelow's [http://math.ucsb.edu/~bigelow/braids.html exploration of B5] Java applet.
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| *C. Nayak and F. Wilczek's connection of projective braid group representations to the fractional quantum Hall effect [http://arxiv.org/abs/cond-mat/9605145]
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| *Presentation for FradkinFest by C. V. Nayak [http://eunahkim.ccmr.cornell.edu/Fradkin-Fest/files/slides/Nayak.ppt]
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| *N. Read's criticism of the reality of Wilczek-Nayak representation [http://arxiv.org/abs/hep-th/0201240]
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| *[http://research.cyber.ee/~lipmaa/crypto//link/public/braid/ Cryptography and Braid Groups page] - Helger Lipmaa
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| *[http://xstructure.inr.ac.ru/x-bin/auththeme3.py?level=1&index1=-63234&skip=0 Braid group: List of Authority Articles on arxiv.org].
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| {{DEFAULTSORT:Braid Group}}
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| [[Category:Braid groups| ]]
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| [[Category:Knot theory]]
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| [[Category:Group theory]]
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| [[Category:Diagram algebras]]
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| [[fr:Tresses (mathématiques)]]
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