Braid group

<h2 id="introduction">Introduction</h2>
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 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:

<table><tbody><tr><td></td><td>   is different from   </td><td></td></tr></tbody></table>
On the other hand, two such connections which can be made to look the same by "pulling the strands" are considered the same braid:

<table><tbody><tr><td></td><td>    is the same as   </td><td></td></tr></tbody></table>
All strands are required to move from left to right; knots like the following are not considered braids:

<table><tbody><tr><td></td><td>   is not a braid</td></tr></tbody></table>
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:

<table><tbody><tr><td></td><td>    composed with    </td><td></td><td>    yields    </td><td></td></tr></tbody></table>
Another example:

<table><tbody><tr><td></td><td>    composed with    </td><td></td><td>    yields    </td><td></td></tr></tbody></table>
The composition of the braids σ and τ is written as στ.
The set of all braids on four strands is denoted by 
 
 
 
 
 B
 
 4
 
 
 
 
 {\displaystyle B_{4}}
 
. The above composition of braids is indeed a <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> operation. The <a href="/facts/Identity_element/9nss3xHY">identity element</a> is the braid consisting of four parallel horizontal strands, and the <a href="/facts/Inverse_element/CwMygHSj">inverse</a> 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.)

<h2 id="applications">Applications</h2>
Braid theory has recently been applied to <a href="/facts/Fluid_mechanics/NIy4tRfE">fluid mechanics</a>, specifically to the field of <a href="/facts/Chaotic_mixing/KfbdXFMl">chaotic mixing</a> 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 <a href="/facts/Topological_entropy/wAEGfb37">topological entropy</a> of several engineered and naturally occurring fluid systems, via the use of <a href="/facts/Nielsen%E2%80%93Thurston_classification/fNJhA8Rk">Nielsen–Thurston classification</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a><a class="footnote-ref" id="fnref:4" href="#fn:4">4</a><a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>
Another field of intense investigation involving braid groups and related topological concepts in the context of <a href="/facts/Quantum_physics/BRc3Mzgr">quantum physics</a> is in the theory and (conjectured) experimental implementation of the proposed particles <a href="/facts/Anyons/hfgpCFjf">anyons</a>. These may well end up forming the basis for error-corrected <a href="/facts/Quantum_computing/nbbcgmvq">quantum computing</a> and so their abstract study is currently of fundamental importance in <a href="/facts/Quantum_information/Bl9QIvHj">quantum information</a>.

<h2 id="formal-treatment">Formal treatment</h2>
Main article: <a href="/facts/Configuration_space_(mathematics)/qBdhCN3z">Configuration space (mathematics) § Connection to braid groups</a>
To put the above informal discussion of braid groups on firm ground, one needs to use the <a href="/facts/Homotopy/1nWrPxdj">homotopy</a> concept of <a href="/facts/Algebraic_topology/2wA8yrjT">algebraic topology</a>, defining braid groups as <a href="/facts/Fundamental_group/VyYtLqSP">fundamental groups</a> of a <a href="/facts/Configuration_space_(mathematics)/qBdhCN3z">configuration space</a>. 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 <a href="/facts/Manifold/rXPERJtu">manifold</a> 
 
 
 
 X
 
 
 {\displaystyle X}
 
 of dimension at least 2. The <a href="/facts/Symmetric_product_(topology)/UIYkdu2E">symmetric product</a> of 
 
 
 
 n
 
 
 {\displaystyle n}
 
 copies of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 means the quotient of 
 
 
 
 
 X
 
 n
 
 
 
 
 {\displaystyle X^{n}}
 
, the 
 
 
 
 n
 
 
 {\displaystyle n}
 
-fold <a href="/facts/Cartesian_product/BfqBWzdL">Cartesian product</a> of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 by the permutation action of the <a href="/facts/Symmetric_group/RUksK5l5">symmetric group</a> on 
 
 
 
 n
 
 
 {\displaystyle n}
 
 strands operating on the indices of coordinates. That is, an ordered 
 
 
 
 n
 
 
 {\displaystyle n}
 
-tuple is in the same <a href="/facts/Orbit_(group_theory)/Vh9VtUUw">orbit</a> as any other that is a re-ordered version of it.
A path in the 
 
 
 
 n
 
 
 {\displaystyle n}
 
-fold symmetric product is the abstract way of discussing 
 
 
 
 n
 
 
 {\displaystyle n}
 
 points of 
 
 
 
 X
 
 
 {\displaystyle X}
 
, considered as an unordered 
 
 
 
 n
 
 
 {\displaystyle n}
 
-tuple, independently tracing out 
 
 
 
 n
 
 
 {\displaystyle n}
 
 strings. Since we must require that the strings never pass through each other, it is necessary that we pass to the subspace 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 of the symmetric product, of orbits of 
 
 
 
 n
 
 
 {\displaystyle n}
 
-tuples of distinct points. That is, we remove all the subspaces of 
 
 
 
 
 X
 
 n
 
 
 
 
 {\displaystyle X^{n}}
 
 defined by conditions 
 
 
 
 
 x
 
 i
 
 
 =
 
 x
 
 j
 
 
 
 
 {\displaystyle x_{i}=x_{j}}
 
 for all 
 
 
 
 1
 ≤
 i
 <
 j
 ≤
 n
 
 
 {\displaystyle 1\leq i<j\leq n}
 
. This is invariant under the symmetric group, and 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 is the quotient by the symmetric group of the non-excluded 
 
 
 
 n
 
 
 {\displaystyle n}
 
-tuples. Under the dimension condition 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 will be connected.
With this definition, then, we can call the braid group of 
 
 
 
 X
 
 
 {\displaystyle X}
 
 with 
 
 
 
 n
 
 
 {\displaystyle n}
 
 strings the fundamental group of 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 (for any choice of base point – this is well-defined <a href="/facts/Up_to/ydz4ztzX">up to</a> isomorphism). The case where 
 
 
 
 X
 
 
 {\displaystyle X}
 
 is the Euclidean plane is the original one of Artin. In some cases it can be shown that the higher <a href="/facts/Homotopy_group/xjogL95Y">homotopy groups</a> of 
 
 
 
 Y
 
 
 {\displaystyle Y}
 
 are trivial.

<h3>Closed braids</h3>
See also: <a href="/facts/Brunnian_braid/pg7S3v0s">Brunnian braid</a>
When X is the plane, the braid can be closed, i.e., corresponding ends can be connected in pairs, to form a <a href="/facts/Link_(knot_theory)/hoFDYmVj">link</a>, 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 <a href="/facts/James_Waddell_Alexander_II/FMPYbH93">J. W. Alexander</a> demonstrates that every link can be obtained in this way as the "closure" of a braid. Compare with <a href="/facts/String_links/hoFDYmVj">string links</a>.
Different braids can give rise to the same link, just as different crossing diagrams can give rise to the same <a href="/facts/Knot_(mathematics)/UMtKJ36m">knot</a>. In 1935, <a href="/facts/Andrey_Markov_Jr./3s1VYSdq">Andrey Markov Jr.</a> described two moves on braid diagrams that yield equivalence in the corresponding closed braids.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> A single-move version of Markov's theorem, was published by in 1997.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>
<a href="/facts/Vaughan_Jones/VMsx3l70">Vaughan Jones</a> originally defined his <a href="/facts/Jones_polynomial/c73vElt9">polynomial</a> as a braid invariant and then showed that it depended only on the class of the closed braid.
The <a href="/facts/Markov_theorem/v2yPKVzA">Markov theorem</a> gives necessary and sufficient conditions under which the closures of two braids are equivalent links.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>

<h3>Braid index</h3>
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 <a href="/facts/Seifert_circle/SledNv2F">Seifert circles</a> in any projection of a knot.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>

<h2 id="history">History</h2>
Braid groups were introduced explicitly by <a href="/facts/Emil_Artin/rI3vIFo2">Emil Artin</a> in 1925, although (as <a href="/facts/Wilhelm_Magnus/WhmRYJL1">Wilhelm Magnus</a> pointed out in 1974<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a>) they were already implicit in <a href="/facts/Adolf_Hurwitz/urNgHh58">Adolf Hurwitz</a>'s work on <a href="/facts/Monodromy/ldkpxExM">monodromy</a> from 1891.
Braid groups may be described by explicit <a href="/facts/Presentation_of_a_group/f9emPUoI">presentations</a>, as was shown by <a href="/facts/Emil_Artin/rI3vIFo2">Emil Artin</a> in 1947.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a> Braid groups are also understood by a deeper mathematical interpretation: as the <a href="/facts/Fundamental_group/VyYtLqSP">fundamental group</a> of certain <a href="/facts/Configuration_space_(mathematics)/qBdhCN3z">configuration spaces</a>.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a>
As Magnus says, Hurwitz gave the interpretation of a braid group as the fundamental group of a configuration space (cf. <a href="/facts/Braid_theory/rn5323Kx">braid theory</a>), an interpretation that was lost from view until it was rediscovered by <a href="/facts/Ralph_Fox/GFlG3J5v">Ralph Fox</a> and Lee Neuwirth in 1962.<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a>

<h2 id="basic-properties">Basic properties</h2>
<h3>Generators and relations</h3>
Consider the following three braids:

<table><tbody><tr><td>      </td><td>      </td><td>      </td></tr><tr><td> σ 1 {\displaystyle \sigma _{1}} </td><td> σ 2 {\displaystyle \sigma _{2}} </td><td> σ 3 {\displaystyle \sigma _{3}} </td></tr></tbody></table>
Every braid in 
 
 
 
 
 B
 
 4
 
 
 
 
 {\displaystyle B_{4}}
 
 can be written as a composition of a number of these braids and their inverses. In other words, these three braids <a href="/facts/Generating_set_of_a_group/B9q5lnFY">generate</a> the group 
 
 
 
 
 B
 
 4
 
 
 
 
 {\displaystyle B_{4}}
 
. 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 
 
 
 
 i
 
 
 {\displaystyle i}
 
 and 
 
 
 
 i
 +
 1
 
 
 {\displaystyle i+1}
 
 is encountered, 
 
 
 
 
 σ
 
 i
 
 
 
 
 {\displaystyle \sigma _{i}}
 
 or 
 
 
 
 
 σ
 
 i
 
 
 −
 1
 
 
 
 
 {\displaystyle \sigma _{i}^{-1}}
 
 is written down, depending on whether strand 
 
 
 
 i
 
 
 {\displaystyle i}
 
 moves over or under strand 
 
 
 
 i
 +
 1
 
 
 {\displaystyle i+1}
 
. Upon reaching the right end, the braid has been written as a product of the 
 
 
 
 
 σ
 
 i
 
 
 
 
 {\displaystyle \sigma _{i}}
 
 and their inverses.
It is clear that

(i) 
 
 
 
 
 σ
 
 1
 
 
 
 σ
 
 3
 
 
 =
 
 σ
 
 3
 
 
 
 σ
 
 1
 
 
 
 
 {\displaystyle \sigma _{1}\sigma _{3}=\sigma _{3}\sigma _{1}}
 
,
while the following two relations are not quite as obvious:

(iia) 
 
 
 
 
 σ
 
 1
 
 
 
 σ
 
 2
 
 
 
 σ
 
 1
 
 
 =
 
 σ
 
 2
 
 
 
 σ
 
 1
 
 
 
 σ
 
 2
 
 
 
 
 {\displaystyle \sigma _{1}\sigma _{2}\sigma _{1}=\sigma _{2}\sigma _{1}\sigma _{2}}
 
,
(iib) 
 
 
 
 
 σ
 
 2
 
 
 
 σ
 
 3
 
 
 
 σ
 
 2
 
 
 =
 
 σ
 
 3
 
 
 
 σ
 
 2
 
 
 
 σ
 
 3
 
 
 
 
 {\displaystyle \sigma _{2}\sigma _{3}\sigma _{2}=\sigma _{3}\sigma _{2}\sigma _{3}}

(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 
 
 
 
 
 σ
 
 1
 
 
 
 
 {\displaystyle \sigma _{1}}
 
, 
 
 
 
 
 σ
 
 2
 
 
 
 
 {\displaystyle \sigma _{2}}
 
 and 
 
 
 
 
 σ
 
 3
 
 
 
 
 {\displaystyle \sigma _{3}}
 
 already follow from these relations and the group axioms.
Generalising this example to 
 
 
 
 n
 
 
 {\displaystyle n}
 
 strands, the group 
 
 
 
 
 B
 
 n
 
 
 
 
 {\displaystyle B_{n}}
 
 can be abstractly defined via the following <a href="/facts/Presentation_of_a_group/f9emPUoI">presentation</a>:

B
          
            n
          
        
        =
        
          ⟨
          
            
              σ
              
                1
              
            
            ,
            …
            ,
            
              σ
              
                n
                −
                1
              
            
            ∣
            
              σ
              
                i
              
            
            
              σ
              
                i
                +
                1
              
            
            
              σ
              
                i
              
            
            =
            
              σ
              
                i
                +
                1
              
            
            
              σ
              
                i
              
            
            
              σ
              
                i
                +
                1
              
            
            ,
            
              σ
              
                i
              
            
            
              σ
              
                j
              
            
            =
            
              σ
              
                j
              
            
            
              σ
              
                i
              
            
          
          ⟩
        
        ,
      
    
    {\displaystyle 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 ,}

where in the first group of relations 
 
 
 
 1
 ≤
 i
 ≤
 n
 −
 2
 
 
 {\displaystyle 1\leq i\leq n-2}
 
 and in the second group of relations 
 
 
 
 
 |
 
 i
 −
 j
 
 |
 
 ≥
 2
 
 
 {\displaystyle |i-j|\geq 2}
 
.<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a><a class="footnote-ref" id="fnref:15" href="#fn:15">15</a> This presentation leads to generalisations of braid groups called <a href="/facts/Artin_group/9BQbTGE8">Artin groups</a>. The cubic relations, known as the braid relations, play an important role in the theory of <a href="/facts/Yang%E2%80%93Baxter_equation/FEzKGDYh">Yang–Baxter equations</a>.

<h3>Further properties</h3>
<ul><li>The braid group 
 
 
 
 
 B
 
 1
 
 
 
 
 {\displaystyle B_{1}}
 
 is <a href="/facts/Trivial_group/XPjQkQFm">trivial</a>, 
 
 
 
 
 B
 
 2
 
 
 
 
 {\displaystyle B_{2}}
 
 is the infinite <a href="/facts/Cyclic_group/JL5zfFuL">cyclic group</a> 
 
 
 
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} }
 
, and 
 
 
 
 
 B
 
 3
 
 
 
 
 {\displaystyle B_{3}}
 
 is isomorphic to the <a href="/facts/Knot_group/bremjfld">knot group</a> of the <a href="/facts/Trefoil_knot/c3K4nUu9">trefoil knot</a> – in particular, it is an infinite <a href="/facts/Abelian_group/FfWwmx4R">non-abelian group</a>.</li>
<li>The n-strand braid group 
 
 
 
 
 B
 
 n
 
 
 
 
 {\displaystyle B_{n}}
 
 embeds as a <a href="/facts/Subgroup/r91mBgpa">subgroup</a> into the 
 
 
 
 (
 n
 +
 1
 )
 
 
 {\displaystyle (n+1)}
 
-strand braid group 
 
 
 
 
 B
 
 n
 +
 1
 
 
 
 
 {\displaystyle B_{n+1}}
 
 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
 
 
 {\displaystyle n\geq 1}
 
 is the infinite braid group 
 
 
 
 
 B
 
 ∞
 
 
 
 
 {\displaystyle B_{\infty }}
 
.</li>
<li>All non-identity elements of 
 
 
 
 
 B
 
 n
 
 
 
 
 {\displaystyle B_{n}}
 
 have infinite <a href="/facts/Order_(group_theory)/Cl3tKGFg">order</a>; i.e., 
 
 
 
 
 B
 
 n
 
 
 
 
 {\displaystyle B_{n}}
 
 is <a href="/facts/Torsion_(algebra)/uNRnwtLz">torsion-free</a>.</li>
<li>There is a left-invariant <a href="/facts/Linear_order/xnTtmuYJ">linear order</a> on 
 
 
 
 
 B
 
 n
 
 
 
 
 {\displaystyle B_{n}}
 
 called the <a href="/facts/Dehornoy_order/s0KYWFgX">Dehornoy order</a>.</li>
<li>For 
 
 
 
 n
 ≥
 3
 
 
 {\displaystyle n\geq 3}
 
, 
 
 
 
 
 B
 
 n
 
 
 
 
 {\displaystyle B_{n}}
 
 contains a subgroup isomorphic to the <a href="/facts/Free_group/zPo2NIGb">free group</a> on two generators.</li>
<li>There is a <a href="/facts/Group_homomorphism/sDguaNYs">homomorphism</a> 
 
 
 
 
 B
 
 n
 
 
 →
 
 Z
 
 
 
 {\displaystyle B_{n}\to \mathbb {Z} }
 
 defined by σi ↦ 1. So for instance, the braid σ2σ3σ1−1σ2σ3 is mapped to 1 + 1 − 1 + 1 + 1 = 3. This map corresponds to the <a href="/facts/Abelianization/oK9WphdE">abelianization</a> of the braid group. Since σik ↦ k, then σik is the identity if and only if 
 
 
 
 k
 =
 0
 
 
 {\displaystyle k=0}
 
. This proves that the generators have infinite order.</li></ul>
<h2 id="interactions">Interactions</h2>
<h3>Relation with symmetric group and the pure braid group</h3>
By forgetting how the strands twist and cross, every braid on n strands determines a <a href="/facts/Permutation/JSw9ukmv">permutation</a> on n elements. This assignment is onto and compatible with composition, and therefore becomes a <a href="/facts/Surjective/KezhLpCb">surjective</a> <a href="/facts/Group_homomorphism/sDguaNYs">group homomorphism</a> Bn → Sn from the braid group onto the <a href="/facts/Symmetric_group/RUksK5l5">symmetric group</a>. The image of the braid σi ∈ Bn is the transposition si = (i, i+1) ∈ Sn. 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 <a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter presentation</a> of the symmetric group:

S
          
            n
          
        
        =
        
          ⟨
          
            
              s
              
                1
              
            
            ,
            …
            ,
            
              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
              
            
            
               for 
            
            
              |
            
            i
            −
            j
            
              |
            
            ≥
            2
            ,
            
              s
              
                i
              
              
                2
              
            
            =
            1
          
          ⟩
        
        .
      
    
    {\displaystyle 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 .}

The <a href="/facts/Kernel_(algebra)/GGSpUPMJ">kernel</a> of the homomorphism Bn → Sn is the subgroup of Bn called the pure braid group on n strands and denoted Pn. This can be seen as the fundamental group of the space of 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 <a href="/facts/Short_exact_sequence/leUJnwJb">short exact sequence</a>

1
        →
        
          F
          
            n
            −
            1
          
        
        →
        
          P
          
            n
          
        
        →
        
          P
          
            n
            −
            1
          
        
        →
        1.
      
    
    {\displaystyle 1\to F_{n-1}\to P_{n}\to P_{n-1}\to 1.}

This sequence splits and therefore pure braid groups are realized as iterated <a href="/facts/Semidirect_product/Uqf87k1I">semi-direct products</a> of free groups.

<h3>Relation between B3 and the modular group</h3>

The braid group 
 
 
 
 
 B
 
 3
 
 
 
 
 {\displaystyle B_{3}}
 
 is the <a href="/facts/Universal_central_extension/t4YguWtF">universal central extension</a> of the <a href="/facts/Modular_group/wL6l14Hk">modular group</a> 
 
 
 
 
 P
 S
 L
 
 (
 2
 ,
 
 Z
 
 )
 
 
 {\displaystyle \mathrm {PSL} (2,\mathbb {Z} )}
 
, with these sitting as lattices inside the (topological) universal covering group

S
 L
 
 (
 2
 ,
 
 R
 
 )
 
 ¯
 
 
 →
 
 P
 S
 L
 
 (
 2
 ,
 
 R
 
 )
 
 
 {\displaystyle {\overline {\mathrm {SL} (2,\mathbb {R} )}}\to \mathrm {PSL} (2,\mathbb {R} )}
 
.
Furthermore, the modular group has trivial center, and thus the modular group is isomorphic to the <a href="/facts/Quotient_group/HlwpHtjb">quotient group</a> of 
 
 
 
 
 B
 
 3
 
 
 
 
 {\displaystyle B_{3}}
 
 modulo its <a href="/facts/Center_(group_theory)/gqAg6B8I">center</a>, 
 
 
 
 Z
 (
 
 B
 
 3
 
 
 )
 ,
 
 
 {\displaystyle Z(B_{3}),}
 
 and equivalently, to the group of <a href="/facts/Inner_automorphism/awJPzhDa">inner automorphisms</a> of 
 
 
 
 
 B
 
 3
 
 
 
 
 {\displaystyle B_{3}}
 
.
Here is a construction of this <a href="/facts/Isomorphic/pj8KgkxU">isomorphism</a>. Define

a
 =
 
 σ
 
 1
 
 
 
 σ
 
 2
 
 
 
 σ
 
 1
 
 
 ,
 
 b
 =
 
 σ
 
 1
 
 
 
 σ
 
 2
 
 
 
 
 {\displaystyle a=\sigma _{1}\sigma _{2}\sigma _{1},\quad b=\sigma _{1}\sigma _{2}}
 
.
From the braid relations it follows that 
 
 
 
 
 a
 
 2
 
 
 =
 
 b
 
 3
 
 
 
 
 {\displaystyle a^{2}=b^{3}}
 
. Denoting this latter product as 
 
 
 
 c
 
 
 {\displaystyle c}
 
, one may verify from the braid relations that

σ
          
            1
          
        
        c
        
          σ
          
            1
          
          
            −
            1
          
        
        =
        
          σ
          
            2
          
        
        c
        
          σ
          
            2
          
          
            −
            1
          
        
        =
        c
      
    
    {\displaystyle \sigma _{1}c\sigma _{1}^{-1}=\sigma _{2}c\sigma _{2}^{-1}=c}

implying that 
 
 
 
 c
 
 
 {\displaystyle c}
 
 is in the center of 
 
 
 
 
 B
 
 3
 
 
 
 
 {\displaystyle B_{3}}
 
. Let 
 
 
 
 C
 
 
 {\displaystyle C}
 
 denote the <a href="/facts/Subgroup/r91mBgpa">subgroup</a> of 
 
 
 
 
 B
 
 3
 
 
 
 
 {\displaystyle B_{3}}
 
 <a href="/facts/Group_generator/B9q5lnFY">generated</a> by c, since C ⊂ Z(B3), it is a <a href="/facts/Normal_subgroup/aRDaGwLm">normal subgroup</a> and one may take the <a href="/facts/Quotient_group/HlwpHtjb">quotient group</a> B3/C. We claim B3/C ≅ PSL(2, Z); this isomorphism can be given an explicit form. The <a href="/facts/Coset/zFgQUTLY">cosets</a> σ1C and σ2C map to

σ
          
            1
          
        
        C
        ↦
        R
        =
        
          
            [
            
              
                
                  1
                
                
                  1
                
              
              
                
                  0
                
                
                  1
                
              
            
            ]
          
        
        
        
          σ
          
            2
          
        
        C
        ↦
        
          L
          
            −
            1
          
        
        =
        
          
            [
            
              
                
                  1
                
                
                  0
                
              
              
                
                  −
                  1
                
                
                  1
                
              
            
            ]
          
        
      
    
    {\displaystyle \sigma _{1}C\mapsto R={\begin{bmatrix}1&1\\0&1\end{bmatrix}}\qquad \sigma _{2}C\mapsto L^{-1}={\begin{bmatrix}1&0\\-1&1\end{bmatrix}}}

where L and R are the standard left and right moves on the <a href="/facts/Stern%E2%80%93Brocot_tree/arS5B10E">Stern–Brocot tree</a>; it is well known that these moves generate the modular group.
Alternately, one common <a href="/facts/Presentation_of_a_group/f9emPUoI">presentation</a> for the modular group is

⟨
        v
        ,
        p
        
        
          |
        
        
        
          v
          
            2
          
        
        =
        
          p
          
            3
          
        
        =
        1
        ⟩
      
    
    {\displaystyle \langle v,p\,|\,v^{2}=p^{3}=1\rangle }

where

v
        =
        
          
            [
            
              
                
                  0
                
                
                  1
                
              
              
                
                  −
                  1
                
                
                  0
                
              
            
            ]
          
        
        ,
        
        p
        =
        
          
            [
            
              
                
                  0
                
                
                  1
                
              
              
                
                  −
                  1
                
                
                  1
                
              
            
            ]
          
        
        .
      
    
    {\displaystyle v={\begin{bmatrix}0&1\\-1&0\end{bmatrix}},\qquad p={\begin{bmatrix}0&1\\-1&1\end{bmatrix}}.}

Mapping a to v and b to p yields a surjective group homomorphism B3 → PSL(2, Z).
The center of B3 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 <a href="/facts/Kernel_(algebra)/GGSpUPMJ">kernel</a> C.

<h3>Relationship to the mapping class group and classification of braids</h3>
The braid group Bn can be shown to be isomorphic to the <a href="/facts/Mapping_class_group/buQm001q">mapping class group</a> of a <a href="/facts/Punctured_disk/rhxKuoem">punctured disk</a> 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 <a href="/facts/Nielsen%E2%80%93Thurston_classification/fNJhA8Rk">periodic, reducible or pseudo-Anosov</a>.

<h3>Connection to knot theory</h3>
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 <a href="/facts/Link_(knot_theory)/hoFDYmVj">link</a>, and sometimes a <a href="/facts/Knot_(mathematics)/UMtKJ36m">knot</a>. <a href="/facts/Alexander%27s_theorem/9cMJHqLu">Alexander's theorem</a> in <a href="/facts/Braid_theory/rn5323Kx">braid theory</a> states that the converse is true as well: every <a href="/facts/Knot_(mathematics)/UMtKJ36m">knot</a> and every <a href="/facts/Link_(knot_theory)/hoFDYmVj">link</a> 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 σi, this is often the preferred method of entering knots into computer programs.

<h3>Computational aspects</h3>
The <a href="/facts/Word_problem_for_groups/eKBrkBPv">word problem</a> for the braid relations is efficiently solvable and there exists a <a href="/facts/Normal_form_(abstract_rewriting)/JERru3Fx">normal form</a> for elements of Bn in terms of the generators σ1, ..., σn−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 <a href="/facts/GAP_(computer_algebra_system)/1yL7xaNk">GAP computer algebra system</a> can carry out computations in Bn 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 <a href="/facts/Lawrence%E2%80%93Krammer_representation/gc1qUbar">Lawrence–Krammer representation</a>.
In addition to the word problem, there are several known hard computational problems that could implement braid groups, applications in <a href="/facts/Cryptography/e94PEVau">cryptography</a> have been suggested.<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a>

<h2 id="actions">Actions</h2>
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 n-tuples of objects or on the n-folded <a href="/facts/Tensor_product/7K3pR1ap">tensor product</a> 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 <a href="/facts/Identity_element/9nss3xHY">identity element</a> of G. Then Bn <a href="/facts/Group_action_(mathematics)/Vh9VtUUw">acts</a> on X in the following fashion:

σ
          
            i
          
        
        
          (
          
            
              x
              
                1
              
            
            ,
            …
            ,
            
              x
              
                i
                −
                1
              
            
            ,
            
              x
              
                i
              
            
            ,
            
              x
              
                i
                +
                1
              
            
            ,
            …
            ,
            
              x
              
                n
              
            
          
          )
        
        =
        
          (
          
            
              x
              
                1
              
            
            ,
            …
            ,
            
              x
              
                i
                −
                1
              
            
            ,
            
              x
              
                i
                +
                1
              
            
            ,
            
              x
              
                i
                +
                1
              
              
                −
                1
              
            
            
              x
              
                i
              
            
            
              x
              
                i
                +
                1
              
            
            ,
            
              x
              
                i
                +
                2
              
            
            ,
            …
            ,
            
              x
              
                n
              
            
          
          )
        
        .
      
    
    {\displaystyle \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).}

Thus the elements xi and xi+1 exchange places and, in addition, xi is twisted by the <a href="/facts/Inner_automorphism/awJPzhDa">inner automorphism</a> corresponding to xi+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 Bn on X. As another example, a <a href="/facts/Braided_monoidal_category/u20sctLa">braided monoidal category</a> is a <a href="/facts/Monoidal_category/F2l5oKMD">monoidal category</a> with a braid group action. Such structures play an important role in modern <a href="/facts/Mathematical_physics/sLKg8T1G">mathematical physics</a> and lead to quantum <a href="/facts/Knot_invariant/BEvEEpHQ">knot invariants</a>.

<h3>Representations</h3>
Elements of the braid group Bn can be represented more concretely by matrices. One classical such <a href="/facts/Group_representation/FsuuDagQ">representation</a> is <a href="/facts/Burau_representation/tvhFEeQC">Burau representation</a>, where the matrix entries are single variable <a href="/facts/Laurent_polynomial/ESxjzdmw">Laurent polynomials</a>. It had been a long-standing question whether Burau representation was <a href="/facts/Faithful_representation/20Wz2jC6">faithful</a>, but the answer turned out to be negative for n ≥ 5. More generally, it was a major open problem whether braid groups were <a href="/facts/Linear_group/wVzG0sa3">linear</a>. In 1990, <a href="/facts/Ruth_Lawrence/onYL6Ru9">Ruth Lawrence</a> described a family of more general "Lawrence representations" depending on several parameters. In 1996, <a href="/facts/Chetan_Nayak/7K4clvL1">Chetan Nayak</a> and <a href="/facts/Frank_Wilczek/EnrB8uUn">Frank Wilczek</a> 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 <a href="/facts/Fractional_quantum_hall_effect/bkRtdmxX">fractional quantum hall effect</a>.<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a> Around 2001 <a href="/facts/Stephen_Bigelow/GrIdCdGF">Stephen Bigelow</a> and Daan Krammer independently proved that all braid groups are linear. Their work used the <a href="/facts/Lawrence%E2%80%93Krammer_representation/gc1qUbar">Lawrence–Krammer representation</a> of dimension 
 
 
 
 n
 (
 n
 −
 1
 )
 
 /
 
 2
 
 
 {\displaystyle n(n-1)/2}
 
 depending on the variables q and t. By suitably specializing these variables, the braid group 
 
 
 
 
 B
 
 n
 
 
 
 
 {\displaystyle B_{n}}
 
 may be realized as a subgroup of the <a href="/facts/General_linear_group/B54gNILS">general linear group</a> over the <a href="/facts/Complex_numbers/2w7mqI7e">complex numbers</a>.

<h2 id="infinitely-generated-braid-groups">Infinitely generated braid groups</h2>
There are many ways to generalize this notion to an infinite number of strands. The simplest way is to take the <a href="/facts/Direct_limit/V3JuHnYK">direct limit</a> of braid groups, where the attaching maps 
 
 
 
 f
 :
 
 B
 
 n
 
 
 →
 
 B
 
 n
 +
 1
 
 
 
 
 {\displaystyle f\colon B_{n}\to B_{n+1}}
 
 send the 
 
 
 
 n
 −
 1
 
 
 {\displaystyle n-1}
 
 generators of 
 
 
 
 
 B
 
 n
 
 
 
 
 {\displaystyle B_{n}}
 
 to the first 
 
 
 
 n
 −
 1
 
 
 {\displaystyle n-1}
 
 generators of 
 
 
 
 
 B
 
 n
 +
 1
 
 
 
 
 {\displaystyle B_{n+1}}
 
 (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 <a href="/facts/Topological_space/v2ut7CbK">topologies</a> that can be imposed on the resulting group each of whose <a href="/facts/Complete_metric_space/lsckmU8G">completion</a> yields a different group.<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a> The first is a very tame group and is isomorphic to the <a href="/facts/Mapping_class_group/buQm001q">mapping class group</a> of the infinitely punctured disk—a discrete set of punctures limiting to the boundary of the <a href="/facts/Unit_disk/phrUu7yC">disk</a>.
The second group can be thought of the same as with finite braid groups. Place a strand at each of the points 
 
 
 
 (
 0
 ,
 1
 
 /
 
 n
 )
 
 
 {\displaystyle (0,1/n)}
 
 and the set of all braids—where a braid is defined to be a collection of paths from the points 
 
 
 
 (
 0
 ,
 1
 
 /
 
 n
 ,
 0
 )
 
 
 {\displaystyle (0,1/n,0)}
 
 to the points 
 
 
 
 (
 0
 ,
 1
 
 /
 
 n
 ,
 1
 )
 
 
 {\displaystyle (0,1/n,1)}
 
 so that the function yields a permutation on endpoints—is isomorphic to this wilder group. An interesting fact is that the pure braid group in this group is isomorphic to both the <a href="/facts/Inverse_limit/f6QecRUu">inverse limit</a> of finite pure braid groups 
 
 
 
 
 P
 
 n
 
 
 
 
 {\displaystyle P_{n}}
 
 and to the <a href="/facts/Fundamental_group/VyYtLqSP">fundamental group</a> of the <a href="/facts/Hilbert_cube/5t1btBr7">Hilbert cube</a> minus the set

{
        (
        
          x
          
            i
          
        
        
          )
          
            i
            ∈
            
              N
            
          
        
        ∣
        
          x
          
            i
          
        
        =
        
          x
          
            j
          
        
        
           for some 
        
        i
        ≠
        j
        }
        .
      
    
    {\displaystyle \{(x_{i})_{i\in \mathbb {N} }\mid x_{i}=x_{j}{\text{ for some }}i\neq j\}.}

<h2 id="cohomology">Cohomology</h2>
See also: <a href="/facts/Configuration_space_(mathematics)/qBdhCN3z">Configuration space (mathematics) § Connection to braid groups</a>
The <a href="/facts/Group_cohomology/ncLTIHj4">cohomology of a group</a> 
 
 
 
 G
 
 
 {\displaystyle G}
 
 is defined as the cohomology of the corresponding <a href="/facts/Eilenberg%E2%80%93MacLane_space/mrux6oMJ">Eilenberg–MacLane</a> <a href="/facts/Classifying_space/gGIAmhtL">classifying space</a>, 
 
 
 
 K
 (
 G
 ,
 1
 )
 
 
 {\displaystyle K(G,1)}
 
, which is a <a href="/facts/CW_complex/rN7buLMo">CW complex</a> uniquely determined by 
 
 
 
 G
 
 
 {\displaystyle G}
 
 up to homotopy. A classifying space for the braid group 
 
 
 
 
 B
 
 n
 
 
 
 
 {\displaystyle B_{n}}
 
 is the nth unordered <a href="/facts/Configuration_space_(mathematics)/qBdhCN3z">configuration space</a> of 
 
 
 
 
 
 R
 
 
 2
 
 
 
 
 {\displaystyle \mathbb {R} ^{2}}
 
, that is, the space of all sets of 
 
 
 
 n
 
 
 {\displaystyle n}
 
 distinct unordered points in the plane:<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a>

UConf
 
 n
 
 
 ⁡
 (
 
 
 R
 
 
 2
 
 
 )
 =
 {
 {
 
 u
 
 1
 
 
 ,
 .
 .
 .
 ,
 
 u
 
 n
 
 
 }
 :
 
 u
 
 i
 
 
 ∈
 
 
 R
 
 
 2
 
 
 ,
 
 u
 
 i
 
 
 ≠
 
 u
 
 j
 
 
 
  for 
 
 i
 ≠
 j
 }
 
 
 {\displaystyle \operatorname {UConf} _{n}(\mathbb {R} ^{2})=\{\{u_{1},...,u_{n}\}:u_{i}\in \mathbb {R} ^{2},u_{i}\neq u_{j}{\text{ for }}i\neq j\}}
 
.
So by definition
The calculations for coefficients in 
 
 
 
 
 Z
 
 
 /
 
 2
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} /2\mathbb {Z} }
 
 can be found in Fuks (1970).<a class="footnote-ref" id="fnref:20" href="#fn:20">20</a>
Similarly, a classifying space for the pure braid group 
 
 
 
 
 P
 
 n
 
 
 
 
 {\displaystyle P_{n}}
 
 is 
 
 
 
 
 Conf
 
 n
 
 
 ⁡
 (
 
 
 R
 
 
 2
 
 
 )
 
 
 {\displaystyle \operatorname {Conf} _{n}(\mathbb {R} ^{2})}
 
, the nth ordered <a href="/facts/Configuration_space_(mathematics)/qBdhCN3z">configuration space</a> of 
 
 
 
 
 
 R
 
 
 2
 
 
 
 
 {\displaystyle \mathbb {R} ^{2}}
 
. In 1968 <a href="/facts/Vladimir_Arnold/IMBhyfMS">Vladimir Arnold</a> showed that the integral cohomology of the pure braid group 
 
 
 
 
 P
 
 n
 
 
 
 
 {\displaystyle P_{n}}
 
 is the quotient of the <a href="/facts/Exterior_algebra/rdN4TvpR">exterior algebra</a> generated by the collection of degree-one classes 
 
 
 
 
 ω
 
 i
 j
 
 
 
 
 1
 ≤
 i
 <
 j
 ≤
 n
 
 
 {\displaystyle \omega _{ij}\;\;1\leq i<j\leq n}
 
, subject to the relations<a class="footnote-ref" id="fnref:21" href="#fn:21">21</a>

ω
          
            k
            ,
            ℓ
          
        
        
          ω
          
            ℓ
            ,
            m
          
        
        +
        
          ω
          
            ℓ
            ,
            m
          
        
        
          ω
          
            m
            ,
            k
          
        
        +
        
          ω
          
            m
            ,
            k
          
        
        
          ω
          
            k
            ,
            ℓ
          
        
        =
        0.
      
    
    {\displaystyle \omega _{k,\ell }\omega _{\ell ,m}+\omega _{\ell ,m}\omega _{m,k}+\omega _{m,k}\omega _{k,\ell }=0.}

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Artin%E2%80%93Tits_group/9BQbTGE8">Artin–Tits group</a></li>
<li><a href="/facts/Braided_monoidal_category/u20sctLa">Braided monoidal category</a></li>
<li><a href="/facts/Braided_vector_space/l5uPLm6b">Braided vector space</a></li>
<li><a href="/facts/Braided_Hopf_algebra/lqBap3VL">Braided Hopf algebra</a></li>
<li><a href="/facts/Knot_theory/mb7rtqVP">Knot theory</a></li>
<li><a href="/facts/Non-commutative_cryptography/T11zAMBm">Non-commutative cryptography</a></li>
<li>Spherical braid group</li></ul>

<h2 id="further-reading">Further reading</h2>

<ul><li><a href="/facts/Joan_Birman/ceIIseYm">Birman, Joan</a>; <a href="/facts/Tara_E._Brendle/59tcgI4y">Brendle, Tara E.</a> (26 February 2005), Braids: A Survey, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math.GT/0409205">math.GT/0409205</a>. In Menasco & Thistlethwaite 2005</li>
<li>Carlucci, Lorenzo; <a href="/facts/Patrick_Dehornoy/NbSfUHXR">Dehornoy, Patrick</a>; Weiermann, Andreas (2011), "Unprovability results involving braids", <a href="/facts/Proceedings_of_the_London_Mathematical_Society/jdcf3vhX">Proceedings of the London Mathematical Society</a>, 3, 102 (1): 159–192, <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/0711.3785">0711.3785</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1112%2Fplms%2Fpdq016">10.1112/plms/pdq016</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2747726">2747726</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:16467487">16467487</a></li>
<li>Chernavskii, A.V. (2001) [1994], <a href="https://www.encyclopediaofmath.org/index.php?title=Braid_theory&oldid=1855">"Braid theory"</a>, <a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a></li>
<li><a href="/facts/Pierre_Deligne/wYC8fhfq">Deligne, Pierre</a> (1972), "Les immeubles des groupes de tresses généralisés", <a href="/facts/Inventiones_Mathematicae/bARcL95Y">Inventiones Mathematicae</a>, 17 (4): 273–302, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1972InMat..17..273D">1972InMat..17..273D</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF01406236">10.1007/BF01406236</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0020-9910">0020-9910</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0422673">0422673</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:123680847">123680847</a></li>
<li><a href="/facts/Ralph_Fox/GFlG3J5v">Fox, Ralph</a>; Neuwirth, Lee (1962), "The braid groups", Mathematica Scandinavica, 10: 119–126, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.7146%2Fmath.scand.a-10518">10.7146/math.scand.a-10518</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0150755">0150755</a></li>
<li>Kassel, Christian; <a href="/facts/Vladimir_Turaev/cGQsYMpD">Turaev, Vladimir</a> (2008), <a href="https://books.google.com/books?id=y6Cox3XjdroC">Braid Groups</a>, Springer, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-33841-5</li>
<li><a href="/facts/William_Menasco/AsJtQBzw">Menasco, William</a>; <a href="/facts/Morwen_Thistlethwaite/Xubb8NWE">Thistlethwaite, Morwen</a>, eds. (2005), Handbook of Knot Theory, Elsevier, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-444-51452-3</li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="https://planetmath.org/BraidGroup">"Braid group"</a>. <a href="/facts/PlanetMath/wphPCtLf">PlanetMath</a>.</li>
<li><a href="http://web.stevens.edu/algebraic/downloads.php">CRAG: CRyptography and Groups</a> computation library from the <a href="/facts/Stevens_University/9It1EQ82">Stevens University</a>'s <a href="http://www.acc.stevens.edu">Algebraic Cryptography Center</a></li>
<li>Macauley, M. <a href="https://www.youtube.com/watch?v=6NElZENlMjI">Lecture 1.3: Groups in science, art, and mathematics</a>. Visual Group Theory. Clemson University.</li>
<li><a href="/facts/Stephen_Bigelow/GrIdCdGF">Bigelow, Stephen</a>. <a href="https://web.archive.org/web/20130604224001/http://www.math.ucsb.edu/~bigelow/braids.html">"Exploration of B5 Java applet"</a>. Archived from <a href="http://math.ucsb.edu/~bigelow/braids.html">the original</a> on 4 June 2013. Retrieved 1 November 2007.</li>
<li>Lipmaa, Helger, <a href="https://web.archive.org/web/20090803144521/http://research.cyber.ee/~lipmaa/crypto/link/public/braid/">Cryptography and Braid Groups page</a>, archived from <a href="http://research.cyber.ee/~lipmaa/crypto/link/public/braid/">the original</a> on 3 August 2009</li>
<li>Dalvit, Ester (2015). <a href="https://www.youtube.com/playlist?list=PLyxHTRWELFBoijzAvXTDOLIF5yfkwN4wZ">Braids – the movie</a>.</li>
<li>Scherich, Nancy. <a href="https://www.youtube.com/watch?v=MASNukczu5A">Representations of the Braid Groups</a>. Dance Your PhD. expanded further in <a href="https://www.youtube.com/watch?v=1za9h2S98Wk">Behind the Math of "Dance Your PhD," Part 1: The Braid Groups.</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Weisstein, Eric. "Braid Group". Wolfram Mathworld. <a href="http://mathworld.wolfram.com/BraidGroup.html" target="_blank">http://mathworld.wolfram.com/BraidGroup.html</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Cohen, Daniel; Suciu, Alexander (1997). "The Braid Monodromy of Plane Algebraic Curves and Hyperplane Arrangements". Commentarii Mathematici Helvetici. 72 (2): 285–315. arXiv:alg-geom/9608001. doi:10.1007/s000140050017. S2CID 14502859. <a href="/wiki/Commentarii_Mathematici_Helvetici" target="_blank">/wiki/Commentarii_Mathematici_Helvetici</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">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, hdl:2142/112556, MR 1742169, S2CID 47710742, archived from the original (PDF) on 26 July 2011 <a href="https://web.archive.org/web/20110726092237/http://www.math.ufl.edu/~boyland/stir.pdf" target="_blank">https://web.archive.org/web/20110726092237/http://www.math.ufl.edu/~boyland/stir.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Gouillart, Emmanuelle; Thiffeault, Jean-Luc; 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, PMID 16605655, S2CID 7142834 <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Stremler, Mark A.; Ross, Shane D.; Grover, Piyush; Kumar, Pankaj (2011), "Topological chaos and periodic braiding of almost-cyclic sets", Physical Review Letters, 106 (11): 114101, Bibcode:2011PhRvL.106k4101S, doi:10.1103/PhysRevLett.106.114101, hdl:10919/24513, PMID 21469863 <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">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 <a href="/wiki/Andrey_Markov_Jr." target="_blank">/wiki/Andrey_Markov_Jr.</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Lambropoulou, Sofia; Rourke, Colin P. (1997), "Markov's theorem in 3-manifolds", Topology and Its Applications, 78 (1–2): 95–122, arXiv:math/0405498, doi:10.1016/S0166-8641(96)00151-4, MR 1465027, S2CID 14494095 <a href="/wiki/Colin_P._Rourke" target="_blank">/wiki/Colin_P._Rourke</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
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</ol>

Braid group open-in-new

Braid group