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changes from revision #67 to #68: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='group_theory'>Group Theory</h4> <div class='hide'> <a class='existingWikiWord' href='/nlab/show/diff/group+theory'>group theory</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/group'>group</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinity-group'>∞-group</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/group+object'>group object</a>, <a class='existingWikiWord' href='/nlab/show/diff/groupoid+object+in+an+%28infinity%2C1%29-category'>group object in an (∞,1)-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/abelian+group'>abelian group</a>, <a class='existingWikiWord' href='/nlab/show/diff/spectrum'>spectrum</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/super+abelian+group'>super abelian group</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/action'>group action</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinity-action'>∞-action</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/representation'>representation</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinity-representation'>∞-representation</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/progroup'>progroup</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/homogeneous+space'>homogeneous space</a></li> </ul> Classical groups <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/general+linear+group'>general linear group</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/unitary+group'>unitary group</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/special+unitary+group'>special unitary group</a>. <a class='existingWikiWord' href='/nlab/show/diff/projective+unitary+group'>projective unitary group</a></li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/orthogonal+group'>orthogonal group</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/special+orthogonal+group'>special orthogonal group</a></li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/symplectic+group'>symplectic group</a> </li> </ul> Finite groups <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/finite+group'>finite group</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/symmetric+group'>symmetric group</a>, <a class='existingWikiWord' href='/nlab/show/diff/cyclic+group'>cyclic group</a>, <a class='existingWikiWord' href='/nlab/show/diff/braid+group'>braid group</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/classification+of+finite+simple+groups'>classification of finite simple groups</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/sporadic+finite+simple+group'>sporadic finite simple groups</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Monster+group'>Monster group</a>, <a class='existingWikiWord' href='/nlab/show/diff/Mathieu+group'>Mathieu group</a></li> </ul> </li> </ul> Group schemes <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/algebraic+group'>algebraic group</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/abelian+variety'>abelian variety</a> </li> </ul> Topological groups <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/topological+group'>topological group</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/compact+topological+group'>compact topological group</a>, <a class='existingWikiWord' href='/nlab/show/diff/locally+compact+topological+group'>locally compact topological group</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/maximal+compact+subgroup'>maximal compact subgroup</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/string+group'>string group</a> </li> </ul> Lie groups <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Lie+group'>Lie group</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/compact+Lie+group'>compact Lie group</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Kac-Moody+group'>Kac-Moody group</a> </li> </ul> Super-Lie groups <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/supergroup'>super Lie group</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/super+Euclidean+group'>super Euclidean group</a> </li> </ul> Higher groups <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/2-group'>2-group</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/crossed+module'>crossed module</a>, <a class='existingWikiWord' href='/nlab/show/diff/strict+2-group'>strict 2-group</a></li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/n-group'>n-group</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/infinity-group'>∞-group</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/simplicial+group'>simplicial group</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/crossed+complex'>crossed complex</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/k-tuply+groupal+n-groupoid'>k-tuply groupal n-groupoid</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/spectrum'>spectrum</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/circle+n-group'>circle n-group</a>, <a class='existingWikiWord' href='/nlab/show/diff/string+2-group'>string 2-group</a>, <a class='existingWikiWord' href='/nlab/show/diff/fivebrane+6-group'>fivebrane Lie 6-group</a> </li> </ul> Cohomology and Extensions <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/group+cohomology'>group cohomology</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/group+extension'>group extension</a>, </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/infinity-group+extension'>∞-group extension</a>, <a class='existingWikiWord' href='/nlab/show/diff/Ext'>Ext-group</a> </li> </ul> Related concepts <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/quantum+group'>quantum group</a></li> </ul> </div> <h4 id='manifolds_and_cobordisms'>Manifolds and cobordisms</h4> <div class='hide'> <a class='existingWikiWord' href='/nlab/show/diff/manifold'>manifolds</a> and <a class='existingWikiWord' href='/nlab/show/diff/cobordism'>cobordisms</a> <a class='existingWikiWord' href='/nlab/show/diff/cobordism+theory'>cobordism theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Cobordism+and+Complex+Oriented+Cohomology'>Introduction</a> Definitions <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/topological+manifold'>locally Euclidean space</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/coordinate+system'>coordinate chart</a>, <a class='existingWikiWord' href='/nlab/show/diff/gluing+function'>coordinate transformation</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/atlas'>atlas</a>, </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/smooth+structure'>smooth structure</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/manifold'>manifold</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/topological+manifold'>topological manifold</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/differentiable+manifold'>differentiable manifold</a>, ,<a class='existingWikiWord' href='/nlab/show/diff/smooth+manifold'>smooth manifold</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/infinite-dimensional+manifold'>infinite dimensional manifold</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Banach+manifold'>Banach manifold</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hilbert+manifold'>Hilbert manifold</a>, <a class='existingWikiWord' href='/nlab/show/diff/ILH+manifold'>ILH manifold</a>, <a class='existingWikiWord' href='/nlab/show/diff/Fr%C3%A9chet+manifold'>Frechet manifold</a>, <a class='existingWikiWord' href='/nlab/show/diff/convenient+manifold'>convenient manifold</a></li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/tangent+bundle'>tangent bundle</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/normal+bundle'>normal bundle</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/G-structure'>G-structure</a>, <a class='existingWikiWord' href='/nlab/show/diff/torsion+of+a+G-structure'>torsion of a G-structure</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/orientation'>orientation</a>, <a class='existingWikiWord' href='/nlab/show/diff/spin+structure'>spin structure</a>, <a class='existingWikiWord' href='/nlab/show/diff/string+structure'>string structure</a>, <a class='existingWikiWord' href='/nlab/show/diff/Fivebrane+structure'>fivebrane structure</a></li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Cartan+geometry'>Cartan geometry</a>: <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Riemannian+manifold'>Riemannian manifold</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/complex+manifold'>complex manifold</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/symplectic+manifold'>symplectic manifold</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/cobordism'>cobordism</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/B-bordism'>B-bordism</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/extended+cobordism'>extended cobordism</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/cobordism+category'>cobordism category</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2Cn%29-category+of+cobordisms'>(∞,n)-category of cobordisms</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/functorial+field+theory'>functorial quantum field theory</a></li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Thom+spectrum'>Thom spectrum</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/bordism+ring'>cobordism ring</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/genus'>genus</a> </li> </ul> </li> </ul> </li> </ul> Genera and invariants <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/signature+genus'>signature genus</a>, <a class='existingWikiWord' href='/nlab/show/diff/Kervaire+invariant'>Kervaire invariant</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/A-hat+genus'>A-hat genus</a>, <a class='existingWikiWord' href='/nlab/show/diff/Witten+genus'>Witten genus</a> </li> </ul> Classification <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/surface'>2-manifolds</a>/<a class='existingWikiWord' href='/nlab/show/diff/surface'>surfaces</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/genus+of+a+surface'>genus of a surface</a></li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/3-manifold'>3-manifolds</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Kirby+calculus'>Kirby calculus</a></li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/4-manifold'>4-manifolds</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Dehn+surgery'>Dehn surgery</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/exotic+smooth+structure'>exotic smooth structure</a> </li> </ul> </li> </ul> Theorems <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Whitney+embedding+theorem'>Whitney embedding theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Thom%27s+transversality+theorem'>Thom's transversality theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Pontrjagin-Thom+collapse+map'>Pontrjagin-Thom construction</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/cobordism+category'>Galatius-Tillmann-Madsen-Weiss theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/geometrization+conjecture'>geometrization conjecture</a>, <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Poincar%C3%A9+conjecture'>Poincaré conjecture</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/elliptization+conjecture'>elliptization conjecture</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/cobordism+hypothesis'>cobordism hypothesis</a>-theorem </li> </ul> </div> <h4 id='knot_theory'>Knot theory</h4> <div class='hide'> <a class='existingWikiWord' href='/nlab/show/diff/knot'>knot theory</a> <a class='existingWikiWord' href='/nlab/show/diff/knot'>knot</a>, <a class='existingWikiWord' href='/nlab/show/diff/link'>link</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/isotopy'>isotopy</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/knot+complement'>knot complement</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/link+diagram'>knot diagrams</a>, <a class='existingWikiWord' href='/nlab/show/diff/chord+diagram'>chord diagram</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Reidemeister+move'>Reidemeister move</a> </li> </ul> Examples/classes: <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/trefoil+knot'>trefoil knot</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/torus+knot'>torus knot</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/singular+knot'>singular knot</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/hyperbolic+link'>hyperbolic knot</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Borromean+link'>Borromean link</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Whitehead+link'>Whitehead link</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Hopf+link'>Hopf link</a> </li> </ul> Types <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/prime+knot'>prime knot</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/mutant+knot'>mutant knot</a> </li> </ul> <a class='existingWikiWord' href='/nlab/show/diff/knot+invariant'>knot invariants</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/crossing+number'>crossing number</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/bridge+number'>bridge number</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/unknotting+number'>unknotting number</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/colorable+knot'>colorability</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/knot+group'>knot group</a> </li> <li> knot genus<a href='/nlab/new/knot+genus'>?</a> </li> <li> polynomial knot invariants (<a class='existingWikiWord' href='/nlab/show/diff/quantum+observable'>observables</a> of <a class='existingWikiWord' href='/nlab/show/diff/non-perturbative+quantum+field+theory'>non-perturbative</a> <a class='existingWikiWord' href='/nlab/show/diff/Chern-Simons+theory'>Chern-Simons theory</a>) <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Jones+polynomial'>Jones polynomial</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/HOMFLY-PT+polynomial'>HOMFLY polynomial</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Alexander+polynomial'>Alexander polynomial</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Reshetikhin-Turaev+construction'>Reshetikhin-Turaev invariants</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Vassiliev+invariant'>Vassiliev knot invariants</a> (<a class='existingWikiWord' href='/nlab/show/diff/quantum+observable'>observables</a> of <a class='existingWikiWord' href='/nlab/show/diff/perturbative+quantum+field+theory'>pertrubative</a> <a class='existingWikiWord' href='/nlab/show/diff/Chern-Simons+theory'>Chern-Simons theory</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Khovanov+homology'>Khovanov homology</a> </li> <li> Kauffman bracket<a href='/nlab/new/Kauffman+bracket'>?</a> </li> </ul> <a class='existingWikiWord' href='/nlab/show/diff/link+invariant'>link invariants</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Milnor+mu-bar+invariant'>Milnor mu-bar invariants</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/linking+number'>linking number</a> </li> </ul> Related concepts: <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Vassiliev+skein+relation'>Vassiliev skein relation</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Seifert+surface'>Seifert surface</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/virtual+knot+theory'>virtual knot theory</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Dehn+surgery'>Dehn surgery</a>, <a class='existingWikiWord' href='/nlab/show/diff/Kirby+calculus'>Kirby calculus</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/volume+conjecture'>volume conjecture</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/arithmetic+topology'>arithmetic topology</a> </li> </ul> </div> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#contents'>Contents</a><ul><li><a href='#Idea'>Idea</a></li><li><a href='#DefinitionsAndCharacterizations'>Definitions and characterizations</a><ul><li><a href='#ByGeneratorsAndRelations'>Via generators and relations</a><ul><li><a href='#presentation_of_general_braids'>Presentation of general braids</a></li></ul></li></ul></li><li><a href='#351'>(-.35,-1)</a></li><li><a href='#351_2'>(+.35,-1)</a></li><li><a href='#351_3'>(-.35,-1)</a></li><li><a href='#351_4'>(+.35,-1)</a></li><li><a href='#351_5'>(-.35,-1)</a></li><li><a href='#351_6'>(+.35,-1)</a></li><li><a href='#351_7'>(-.35,-1)</a></li><li><a href='#351_8'>(+.35,-1)</a></li><li><a href='#353'>(-.35,3)</a></li><li><a href='#1055'>(+1.05,5)</a></li></ul></li><li><a href='#'>\;\;\;\;\;</a><ul><li><a href='#1053'>(1.05,3)</a></li><li><a href='#355'>(-.35,5)</a></li><li><a href='#351_9'>(-.35,1)</a><ul><li><a href='#PresentationOfPureBraids'>Presentation of pure braids</a></li></ul></li></ul></li><li><a href='#_2'>\;\;</a><ul><li><a href='#21'>(-2,-1)</a></li><li><a href='#28'>(-2,+.8)</a></li><li><a href='#21_2'>(-2,-1)</a></li><li><a href='#28_2'>(-2,+.8)</a></li><li><a href='#21_3'>(-2,-1)</a></li><li><a href='#28_3'>(-2,+.8)</a></li><li><a href='#21_4'>(-2,-1)</a></li><li><a href='#28_4'>(-2,+.8)</a></li></ul></li><li><a href='#_3'>\;\;\;</a><ul><li><a href='#21_5'>(-2,-1)</a></li><li><a href='#28_5'>(-2,+.8)</a></li><li><a href='#21_6'>(-2,-1)</a></li><li><a href='#28_6'>(-2,+.8)</a></li><li><a href='#21_7'>(-2,-1)</a></li><li><a href='#28_7'>(-2,+.8)</a></li><li><a href='#21_8'>(-2,-1)</a></li><li><a href='#28_8'>(-2,+.8)</a></li><li><a href='#11'>(+1,-1)</a></li><li><a href='#18'>(+1,+.8)</a></li><li><a href='#11_2'>(+1,-1)</a></li><li><a href='#18_2'>(+1,+.8)</a></li><li><a href='#11_3'>(-1,-1)</a></li><li><a href='#1_8'>(-1, +.8)</a></li><li><a href='#11_4'>(-1,-1)</a></li><li><a href='#1_8_2'>(-1, +.8)</a></li></ul></li><li><a href='#_4'>\;\;</a><ul><li><a href='#11_5'>(+1,-1)</a></li><li><a href='#18_3'>(+1,+.8)</a></li><li><a href='#11_6'>(+1,-1)</a></li><li><a href='#18_4'>(+1,+.8)</a></li><li><a href='#11_7'>(-1,-1)</a></li><li><a href='#1_8_3'>(-1, +.8)</a></li><li><a href='#11_8'>(-1,-1)</a></li><li><a href='#1_8_4'>(-1, +.8)</a></li><li><a href='#11_9'>(+1,-1)</a></li><li><a href='#18_5'>(+1,+.8)</a></li><li><a href='#11_10'>(+1,-1)</a></li><li><a href='#18_6'>(+1,+.8)</a></li><li><a href='#11_11'>(-1,-1)</a></li><li><a href='#1_8_5'>(-1, +.8)</a></li><li><a href='#11_12'>(-1,-1)</a></li><li><a href='#1_8_6'>(-1, +.8)</a></li></ul></li><li><a href='#_5'>\;\;</a><ul><li><a href='#11_13'>(+1,-1)</a></li><li><a href='#18_7'>(+1,+.8)</a></li><li><a href='#11_14'>(+1,-1)</a></li><li><a href='#18_8'>(+1,+.8)</a></li><li><a href='#11_15'>(-1,-1)</a></li><li><a href='#1_8_7'>(-1, +.8)</a></li><li><a href='#11_16'>(-1,-1)</a></li><li><a href='#1_8_8'>(-1, +.8)</a></li><li><a href='#1_1'>(+1, -1)</a></li><li><a href='#1_8_9'>(+1, .8)</a></li><li><a href='#1_1_2'>(+1, -1)</a></li><li><a href='#1_8_10'>(+1, .8)</a></li><li><a href='#01'>(0,-1)</a></li><li><a href='#0_8'>(0, +.8)</a></li><li><a href='#01_2'>(0,-1)</a></li><li><a href='#0_8_2'>(0, +.8)</a></li></ul></li><li><a href='#_6'>\;\;\;</a><ul><li><a href='#1_1_3'>(+1, -1)</a></li><li><a href='#1_8_11'>(+1, .8)</a></li><li><a href='#1_1_4'>(+1, -1)</a></li><li><a href='#1_8_12'>(+1, .8)</a></li><li><a href='#01_3'>(0,-1)</a></li><li><a href='#0_8_3'>(0, +.8)</a></li><li><a href='#01_4'>(0,-1)</a></li><li><a href='#0_8_4'>(0, +.8)</a></li><li><a href='#1_1_5'>(+1, -1)</a></li><li><a href='#1_8_13'>(+1, .8)</a></li><li><a href='#1_1_6'>(+1, -1)</a></li><li><a href='#1_8_14'>(+1, .8)</a></li><li><a href='#2_1'>(+2, -1)</a></li><li><a href='#2_8'>(+2, +.8)</a></li><li><a href='#2_1_2'>(+2, -1)</a></li><li><a href='#2_8_2'>(+2, +.8)</a></li></ul></li><li><a href='#_7'>\;\;\;</a><ul><li><a href='#1_1_7'>(+1, -1)</a></li><li><a href='#1_8_15'>(+1, .8)</a></li><li><a href='#1_1_8'>(+1, -1)</a></li><li><a href='#1_8_16'>(+1, .8)</a></li><li><a href='#2_1_3'>(+2, -1)</a></li><li><a href='#2_8_3'>(+2, +.8)</a></li><li><a href='#2_1_4'>(+2, -1)</a></li><li><a href='#2_8_4'>(+2, +.8)</a></li><li><a href='#11_17'>(-1,-1)</a></li><li><a href='#1_8_17'>(-1, .8)</a></li><li><a href='#11_18'>(-1,-1)</a></li><li><a href='#1_8_18'>(-1, .8)</a></li><li><a href='#11_19'>(-1,-1)</a></li><li><a href='#1_8_19'>(-1, .8)</a></li><li><a href='#11_20'>(-1,-1)</a></li><li><a href='#1_8_20'>(-1, .8)</a></li><li><a href='#11_21'>(-1,-1)</a></li><li><a href='#1_8_21'>(-1, .8)</a></li><li><a href='#11_22'>(-1,-1)</a></li><li><a href='#1_8_22'>(-1, .8)</a></li></ul></li><li><a href='#_8'>\;\;\;</a><ul><li><a href='#11_23'>(-1,-1)</a></li><li><a href='#1_8_23'>(-1, .8)</a></li><li><a href='#11_24'>(-1,-1)</a></li><li><a href='#1_8_24'>(-1, .8)</a></li><li><a href='#11_25'>(-1,-1)</a></li><li><a href='#1_8_25'>(-1, .8)</a></li><li><a href='#11_26'>(-1,-1)</a></li><li><a href='#1_8_26'>(-1, .8)</a></li></ul></li><li><a href='#_9'>\;\;\;</a><ul><li><a href='#11_27'>(-1,-1)</a></li><li><a href='#1_8_27'>(-1, .8)</a></li><li><a href='#11_28'>(-1,-1)</a></li><li><a href='#1_8_28'>(-1, .8)</a><ul><li><a href='#AsFundamentalGroupOfConfigurationSpace'>As fundamental group of a configuration space of points</a></li><li><a href='#AsMappingClassGroupOfAPuncturedDisk'>As mapping class group of punctured disk</a></li><li><a href='#AsAutomorphismsOfAFreeGroup'>As automorphisms of a free group</a></li></ul></li><li><a href='#properties'>Properties</a><ul><li><a href='#relation_to_moduli_space_of_monopoles'>Relation to moduli space of monopoles</a></li></ul></li><li><a href='#Examples'>Examples</a></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a><ul><li><a href='#general'>General</a></li><li><a href='#braid_group_representations_as_topological_quantum_gates'>Braid group representations (as topological quantum gates)</a></li><li><a href='#ReferencesGraphBraidGroups'>Graph braid groups</a></li><li><a href='#relation_to_moduli_space_of_monopoles_2'>Relation to moduli space of monopoles</a></li><li><a href='#BraidGroupCryptographyReferences'>Braid group cryptography</a><ul><li><a href='#via_conjugacy_search'>Via Conjugacy Search</a></li><li><a href='#via_emultiplication'>Via E-multiplication</a></li><li><a href='#further_developments'>Further developments</a></li></ul></li></ul></li></ul></li></ul></div> <h2 id='Idea'>Idea</h2> By a braid group [[Artin (1925)](#Artin25)] one means the <a class='existingWikiWord' href='/nlab/show/diff/group'>group</a> of joint continuous motions [cf. Goldsmith (1981)] of a fixed number <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>n+1</annotation></semantics></math> of non-coincident points in the <a class='existingWikiWord' href='/nlab/show/diff/plane'>plane</a>, from any fixed <a class='existingWikiWord' href='/nlab/show/diff/configuration+space+of+points'>configuration</a> back to that fixed configuration. The “<a class='existingWikiWord' href='/nlab/show/diff/worldvolume'>worldlines</a>” traced out by such points in space-time under such an operation look like a braid with <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>n+1</annotation></semantics></math> strands, whence the name. <img src='/nlab/files/GenericBraidGroupElement-230205.jpg' width='280' /> As with actual braids, here it is understood that two such operations are identified if they differ only by continuous deformations of the “strands” without breaking or intersecting these, hence that one identifies those such systems of <a class='existingWikiWord' href='/nlab/show/diff/worldvolume'>worldlines</a> which are <a class='existingWikiWord' href='/nlab/show/diff/isotopy'>isotopic</a> in <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝ</mi> <mn>3</mn></msup></mrow><annotation encoding='application/x-tex'>\mathbb{R}^3</annotation></semantics></math>. (This is just the kind of invariance considered for <a class='existingWikiWord' href='/nlab/show/diff/link+diagram'>link diagrams</a> — such as under <a class='existingWikiWord' href='/nlab/show/diff/Reidemeister+move'>Reidemeister moves</a> — and in fact every link diagram may be obtained by “closing up” a braid diagram, in the evident sense.) A quick way of saying this with precision is to observe that a braid group is thus the <a class='existingWikiWord' href='/nlab/show/diff/fundamental+group'>fundamental group</a> <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>\pi_1</annotation></semantics></math> of a <a class='existingWikiWord' href='/nlab/show/diff/configuration+space+of+points'>configuration space of points</a> in the plane (see <a href='#AsFundamentalGroupOfConfigurationSpace'>below</a> for more). Here it makes a key difference whether: <ul> <li> one considers the points in a configuration as ordered (labeled by numbers <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>1, \cdots, n+1</annotation></semantics></math>) in which case one speaks of the <a class='existingWikiWord' href='/nlab/show/diff/braid+group'>pure braid group</a>, </li> <li> or as indistinguishable (albeit in any case with distinct positions!) in which case one speaks of the braid group proper. </li> </ul> Namely, after traveling along a general braid <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>b</mi></mrow><annotation encoding='application/x-tex'>b</annotation></semantics></math> the order of the given points may come out permuted by a <a class='existingWikiWord' href='/nlab/show/diff/permutation'>permutation</a> <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi mathvariant='normal'>perm</mi><mo stretchy='false'>(</mo><mi>b</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathrm{perm}(b)</annotation></semantics></math>, and the braid is called pure precisely if this permutation is trivial: <img src='/nlab/files/BraidGroupFibration-230205.jpg' width='600' /> More explicitly, braid groups admit <a class='existingWikiWord' href='/nlab/show/diff/finitely+presentable+group'>finite presentations</a> by <a class='existingWikiWord' href='/nlab/show/diff/generators+and+relations'>generators and relations</a>. To these we now turn first: \linebreak <h2 id='DefinitionsAndCharacterizations'>Definitions and characterizations</h2> <ul> <li> <a href='#ByGeneratorsAndRelations'>Via generators and relations</a> </li> <li> <a href='#AsFundamentalGroupOfConfigurationSpace'>As the fundamental group of a configuration space of points</a> </li> <li> <a href='#AsMappingClassGroupOfAPuncturedDisk'>As mapping class group of a punctured disk</a> </li> <li> <a href='#AsAutomorphismsOfAFreeGroup'>As automorphisms of a free group</a> </li> </ul> <h4 id='ByGeneratorsAndRelations'>Via generators and relations</h4> <h5 id='presentation_of_general_braids'>Presentation of general braids</h5> We discuss the braid group as a <a class='existingWikiWord' href='/nlab/show/diff/finitely+generated+group'>finitely generated group</a> (<a href='#Artin25'>Artin 1925, (5)-(6)</a>; <a href='#Artin47'>Artin 1947, (18)-(19)</a>; review in, e.g.: <a href='#FoxNeuwirth62'>Fox & Neuwirth 1962, §7</a>): \begin{definition}\label{ArtinPresentation} (Artin presentation) The Artin braid group, <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Br</mi><mo stretchy='false'>(</mo><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Br({n+1})</annotation></semantics></math>, on <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>n+1</annotation></semantics></math> strands is the <a class='existingWikiWord' href='/nlab/show/diff/finitely+generated+group'>finitely generated group</a> given via <a class='existingWikiWord' href='/nlab/show/diff/generators+and+relations'>generators and relations</a> by (this and the following graphics are taken from <a class='existingWikiWord' href='/schreiber/show/diff/Topological+Quantum+Gates+in+Homotopy+Type+Theory' title='schreiber'>Myers et al. (2023)</a>): <ul> <li> generators: <div class='maruku-equation' id='eq:ArtinGenerators'>(1)<math class='maruku-mathml' display='block' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>b</mi> <mi>i</mi></msub><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>…</mi><mo>,</mo><mi>n</mi></mrow><annotation encoding='application/x-tex'> b_i\;\; i = 1, \ldots, n </annotation></semantics></math></div></li> </ul> \begin{tikzcd} b_i \;:=\; \color{orange} \left[\color{black} \raisebox{6pt}{ \begin{tikzpicture}] \begin{scope}[shift={(-1.6,0)}] \draw[line width=1.4] (-.5,1) to (-.5,-1); \draw[line width=1.2] (+.5,1) to (+.5,-1); \draw (-0,-.1) node {<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mpadded lspace='-50%width' width='0'><mi>⋯</mi></mpadded></mrow><annotation encoding='application/x-tex'>\mathclap{\cdots}</annotation></semantics></math>}; \end{scope} \draw[line width=1.4] (-.35,-1) .. controls (-.35,0) and (+.35,0) .. (+.35,1); \draw[line width=4.5, white] (+.35,-1) .. controls (+.35,0) and (-.35,0) .. (-.35,1); \draw[line width=1.4] (+.35,-1) .. controls (+.35,0) and (-.35,0) .. (-.35,1); \begin{scope}[shift={(+1.6,0)}] \draw[line width=1.4] (-.5,1) to (-.5,-1); \draw[line width=1.2] (+.5,1) to (+.5,-1); \draw (-0,-.1) node {<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mpadded lspace='-50%width' width='0'><mi>⋯</mi></mpadded></mrow><annotation encoding='application/x-tex'>\mathclap{\cdots}</annotation></semantics></math>}; \end{scope} \draw (-2.1, -1.3) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math>}}; \draw (-1.1, -1.3) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mspace width='negativethinmathspace' /><mo>−</mo><mspace width='negativethinmathspace' /><mn>1</mn></mrow><annotation encoding='application/x-tex'>i\!-\!1</annotation></semantics></math>}}; \draw (-.4, -1.3) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi></mrow><annotation encoding='application/x-tex'>i</annotation></semantics></math>}}; \draw (+.35, -1.3) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mspace width='negativethinmathspace' /><mo>+</mo><mspace width='negativethinmathspace' /><mn>1</mn></mrow><annotation encoding='application/x-tex'>i\!+\!1</annotation></semantics></math>}}; \draw (+1.1, -1.3) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mspace width='negativethinmathspace' /><mo>+</mo><mspace width='negativethinmathspace' /><mn>2</mn></mrow><annotation encoding='application/x-tex'>i\!+\!2</annotation></semantics></math>}}; \draw (+2.1, -1.3) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>n+1</annotation></semantics></math>}}; \end{tikzpicture} } \color{orange} \right] \color{black} \end{tikzcd} \begin{tikzcd} b_i^{-1} \;:=\; \color{orange} \left[\color{black} \raisebox{6pt}{ \begin{tikzpicture}] \begin{scope}[shift={(-1.6,0)}] \draw[line width=1.4] (-.5,1) to (-.5,-1); \draw[line width=1.2] (+.5,1) to (+.5,-1); \draw (-0,-.1) node {<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mpadded lspace='-50%width' width='0'><mi>⋯</mi></mpadded></mrow><annotation encoding='application/x-tex'>\mathclap{\cdots}</annotation></semantics></math>}; \end{scope} \draw[line width=1.4] (+.35,-1) .. controls (+.35,0) and (-.35,0) .. (-.35,1); \draw[line width=4.5, white] (-.35,-1) .. controls (-.35,0) and (+.35,0) .. (+.35,1); \draw[line width=1.4] (-.35,-1) .. controls (-.35,0) and (+.35,0) .. (+.35,1); \begin{scope}[shift={(+1.6,0)}] \draw[line width=1.4] (-.5,1) to (-.5,-1); \draw[line width=1.2] (+.5,1) to (+.5,-1); \draw (-0,-.1) node {<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mpadded lspace='-50%width' width='0'><mi>⋯</mi></mpadded></mrow><annotation encoding='application/x-tex'>\mathclap{\cdots}</annotation></semantics></math>}; \end{scope} \draw (-2.1, -1.3) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math>}}; \draw (-1.1, -1.3) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mspace width='negativethinmathspace' /><mo>−</mo><mspace width='negativethinmathspace' /><mn>1</mn></mrow><annotation encoding='application/x-tex'>i\!-\!1</annotation></semantics></math>}}; \draw (-.4, -1.3) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi></mrow><annotation encoding='application/x-tex'>i</annotation></semantics></math>}}; \draw (+.35, -1.3) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mspace width='negativethinmathspace' /><mo>+</mo><mspace width='negativethinmathspace' /><mn>1</mn></mrow><annotation encoding='application/x-tex'>i\!+\!1</annotation></semantics></math>}}; \draw (+1.1, -1.3) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mspace width='negativethinmathspace' /><mo>+</mo><mspace width='negativethinmathspace' /><mn>2</mn></mrow><annotation encoding='application/x-tex'>i\!+\!2</annotation></semantics></math>}}; \draw (+2.1, -1.3) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>n+1</annotation></semantics></math>}}; \end{tikzpicture} } \color{orange} \right] \color{black} \end{tikzcd} <ul> <li> relations I: <div class='maruku-equation' id='eq:ArtinRelations'>(2)<math class='maruku-mathml' display='block' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo>∀</mo><mrow><mi>i</mi><mo>+</mo><mn>1</mn><mo><</mo><mi>j</mi></mrow></munder><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><msub><mi>b</mi> <mi>i</mi></msub><mo>⋅</mo><msub><mi>b</mi> <mi>j</mi></msub><mspace width='thickmathspace' /><mo>=</mo><mspace width='thickmathspace' /><msub><mi>b</mi> <mi>j</mi></msub><mo>⋅</mo><msub><mi>b</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'> \underset{ i+1 \lt j }{\forall} \;\;\; b_i \cdot b_j \;=\; b_j \cdot b_i </annotation></semantics></math></div></li> </ul> \begin{tikzcd} \color{orange} \left[\color{black} \raisebox{-20pt}{ \begin{tikzpicture}] \begin{scope} \begin{scope}[shift={(-1.6,0)}] \draw[line width=1.4] (-.5,1) to (-.5,-1); \draw[line width=1.2] (+.5,1) to (+.5,-1); \draw (-0,-.1) node {<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mpadded lspace='-50%width' width='0'><mi>⋯</mi></mpadded></mrow><annotation encoding='application/x-tex'>\mathclap{\cdots}</annotation></semantics></math>}; \end{scope} \draw[line width=1.4] (-.35,-1) .. controls (-.35,0) and (+.35,0) .. (+.35,1); \draw[line width=4.5, white] (+.35,-1) .. controls (+.35,0) and (-.35,0) .. (-.35,1); \draw[line width=1.4] (+.35,-1) .. controls (+.35,0) and (-.35,0) .. (-.35,1); \begin{scope}[shift={(+1.6,0)}] \draw[line width=1.4] (-.5,1) to (-.5,-1); \draw[line width=1.2] (+.5,1) to (+.5,-1); \draw (-0,-.1) node {<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mpadded lspace='-50%width' width='0'><mi>⋯</mi></mpadded></mrow><annotation encoding='application/x-tex'>\mathclap{\cdots}</annotation></semantics></math>}; \end{scope} \begin{scope}[shift={(3.2,0)}] \draw[line width=1.4] <h2 id='351'>(-.35,-1)</h2> (-.35,1); \draw[line width=1.4] <h2 id='351_2'>(+.35,-1)</h2> (+.35,1); \end{scope} \begin{scope}[shift={(+4.8,0)}] \draw[line width=1.4] (-.5,1) to (-.5,-1); \draw[line width=1.2] (+.5,1) to (+.5,-1); \draw (-0,-.1) node {<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mpadded lspace='-50%width' width='0'><mi>⋯</mi></mpadded></mrow><annotation encoding='application/x-tex'>\mathclap{\cdots}</annotation></semantics></math>}; \end{scope} \draw (-2.1, -1.3) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math>}}; \draw (-1.1, -1.3) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mspace width='negativethinmathspace' /><mo>−</mo><mspace width='negativethinmathspace' /><mn>1</mn></mrow><annotation encoding='application/x-tex'>i\!-\!1</annotation></semantics></math>}}; \draw (-.4, -1.3) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi></mrow><annotation encoding='application/x-tex'>i</annotation></semantics></math>}}; \draw (+.35, -1.3) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mspace width='negativethinmathspace' /><mo>+</mo><mspace width='negativethinmathspace' /><mn>1</mn></mrow><annotation encoding='application/x-tex'>i\!+\!1</annotation></semantics></math>}}; \draw (+1.1, -1.3) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mspace width='negativethinmathspace' /><mo>+</mo><mspace width='negativethinmathspace' /><mn>2</mn></mrow><annotation encoding='application/x-tex'>i\!+\!2</annotation></semantics></math>}}; \draw (+2.1, -1.3) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>j</mi><mspace width='negativethinmathspace' /><mo>−</mo><mspace width='negativethinmathspace' /><mn>1</mn></mrow><annotation encoding='application/x-tex'>j\!-\!1</annotation></semantics></math>}}; \draw (+2.8, -1.3) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>j</mi></mrow><annotation encoding='application/x-tex'>j</annotation></semantics></math>}}; \draw (+3.5, -1.3) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>j</mi><mspace width='negativethinmathspace' /><mo>+</mo><mspace width='negativethinmathspace' /><mn>1</mn></mrow><annotation encoding='application/x-tex'>j\!+\!1</annotation></semantics></math>}}; \draw (+4.3, -1.3) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>j</mi><mspace width='negativethinmathspace' /><mo>+</mo><mspace width='negativethinmathspace' /><mn>2</mn></mrow><annotation encoding='application/x-tex'>j\!+\!2</annotation></semantics></math>}}; \draw (+5.3, -1.3) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mspace width='negativethinmathspace' /><mo>+</mo><mspace width='negativethinmathspace' /><mn>1</mn></mrow><annotation encoding='application/x-tex'>n\!+\!1</annotation></semantics></math>}}; \end{scope} \begin{scope}[shift={(0,2)}] \begin{scope}[shift={(-1.6,0)}] \draw[line width=1.4] (-.5,1) to (-.5,-1); \draw[line width=1.2] (+.5,1) to (+.5,-1); \draw (-0,-.1) node {<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mpadded lspace='-50%width' width='0'><mi>⋯</mi></mpadded></mrow><annotation encoding='application/x-tex'>\mathclap{\cdots}</annotation></semantics></math>}; \end{scope} \begin{scope} \draw[line width=1.4] <h2 id='351_3'>(-.35,-1)</h2> (-.35,1); \draw[line width=1.4] <h2 id='351_4'>(+.35,-1)</h2> (+.35,1); \end{scope} \begin{scope}[shift={(+1.6,0)}] \draw[line width=1.4] (-.5,1) to (-.5,-1); \draw[line width=1.2] (+.5,1) to (+.5,-1); \draw (-0,-.1) node {<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mpadded lspace='-50%width' width='0'><mi>⋯</mi></mpadded></mrow><annotation encoding='application/x-tex'>\mathclap{\cdots}</annotation></semantics></math>}; \end{scope} \begin{scope}[shift={(3.2,0)}] \draw[line width=1.4] (-.35,-1) .. controls (-.35,0) and (+.35,0) .. (+.35,1); \draw[line width=4.5, white] (+.35,-1) .. controls (+.35,0) and (-.35,0) .. (-.35,1); \draw[line width=1.4] (+.35,-1) .. controls (+.35,0) and (-.35,0) .. (-.35,1); \end{scope} \begin{scope}[shift={(+4.8,0)}] \draw[line width=1.4] (-.5,1) to (-.5,-1); \draw[line width=1.2] (+.5,1) to (+.5,-1); \draw (-0,-.1) node {<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mpadded lspace='-50%width' width='0'><mi>⋯</mi></mpadded></mrow><annotation encoding='application/x-tex'>\mathclap{\cdots}</annotation></semantics></math>}; \end{scope} \end{scope} \end{tikzpicture} } \color{orange} \right] \color{black} \;\;\;\;\; = \;\;\;\;\; \color{orange} \left[\color{black} \raisebox{-20pt}{ \begin{tikzpicture}] \begin{scope}[shift={(0,2)}] \begin{scope} \begin{scope}[shift={(-1.6,0)}] \draw[line width=1.4] (-.5,1) to (-.5,-1); \draw[line width=1.2] (+.5,1) to (+.5,-1); \draw (-0,-.1) node {<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mpadded lspace='-50%width' width='0'><mi>⋯</mi></mpadded></mrow><annotation encoding='application/x-tex'>\mathclap{\cdots}</annotation></semantics></math>}; \end{scope} \draw[line width=1.4] (-.35,-1) .. controls (-.35,0) and (+.35,0) .. (+.35,1); \draw[line width=4.5, white] (+.35,-1) .. controls (+.35,0) and (-.35,0) .. (-.35,1); \draw[line width=1.4] (+.35,-1) .. controls (+.35,0) and (-.35,0) .. 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(+.35,1); \draw[line width=4.5, white] (+.35,-1) .. controls (+.35,0) and (-.35,0) .. (-.35,1); \draw[line width=1.4] (+.35,-1) .. controls (+.35,0) and (-.35,0) .. (-.35,1); \end{scope} \begin{scope}[shift={(+4.8,0)}] \draw[line width=1.4] (-.5,1) to (-.5,-1); \draw[line width=1.2] (+.5,1) to (+.5,-1); \draw (-0,-.1) node {<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mpadded lspace='-50%width' width='0'><mi>⋯</mi></mpadded></mrow><annotation encoding='application/x-tex'>\mathclap{\cdots}</annotation></semantics></math>}; \end{scope} \draw (-2.1, -1.3) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math>}}; \draw (-1.1, -1.3) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mspace width='negativethinmathspace' /><mo>−</mo><mspace width='negativethinmathspace' /><mn>1</mn></mrow><annotation encoding='application/x-tex'>i\!-\!1</annotation></semantics></math>}}; 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\draw (+2.1, -1.3) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>j</mi><mspace width='negativethinmathspace' /><mo>−</mo><mspace width='negativethinmathspace' /><mn>1</mn></mrow><annotation encoding='application/x-tex'>j\!-\!1</annotation></semantics></math>}}; \draw (+2.8, -1.3) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>j</mi></mrow><annotation encoding='application/x-tex'>j</annotation></semantics></math>}}; \draw (+3.5, -1.3) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>j</mi><mspace width='negativethinmathspace' /><mo>+</mo><mspace width='negativethinmathspace' /><mn>1</mn></mrow><annotation encoding='application/x-tex'>j\!+\!1</annotation></semantics></math>}}; \draw (+4.3, -1.3) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>j</mi><mspace width='negativethinmathspace' /><mo>+</mo><mspace width='negativethinmathspace' /><mn>2</mn></mrow><annotation encoding='application/x-tex'>j\!+\!2</annotation></semantics></math>}}; \draw (+5.3, -1.3) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mspace width='negativethinmathspace' /><mo>+</mo><mspace width='negativethinmathspace' /><mn>1</mn></mrow><annotation encoding='application/x-tex'>n\!+\!1</annotation></semantics></math>}}; \end{scope} \end{tikzpicture} } \color{orange} \right] \color{black} \end{tikzcd} \linebreak <ul> <li> relations II <div class='maruku-equation' id='eq:eq3'>(3)<math class='maruku-mathml' display='block' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo>∀</mo><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo><</mo><mi>n</mi></mrow></munder><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><msub><mi>b</mi> <mi>i</mi></msub><mo>⋅</mo><msub><mi>b</mi> <mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>⋅</mo><msub><mi>b</mi> <mi>i</mi></msub><mspace width='thickmathspace' /><mo>=</mo><mspace width='thickmathspace' /><msub><mi>b</mi> <mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>⋅</mo><msub><mi>b</mi> <mi>i</mi></msub><mo>⋅</mo><msub><mi>b</mi> <mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding='application/x-tex'> \underset{ 1 \leq i \lt n }{\forall} \;\;\; b_i \cdot b_{i+1} \cdot b_i \;=\; b_{i+1} \cdot b_i \cdot b_{i+1} </annotation></semantics></math></div></li> </ul> \begin{tikzcd} \color{orange} \left[\color{black} \raisebox{-24pt}{ \begin{tikzpicture}yscale=.5] \begin{scope}[shift={(-1.6,0)}] \draw[line width=1.4] (-.5,5) to (-.5,-1); \draw[line width=1.2] (+.5,5) to (+.5,-1); \draw (-0,2.2) node {<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mpadded lspace='-50%width' width='0'><mi>⋯</mi></mpadded></mrow><annotation encoding='application/x-tex'>\mathclap{\cdots}</annotation></semantics></math>}; \end{scope} \begin{scope}[shift={(0,4)}] \draw[line width=1.4] (-.35,-1) .. controls (-.35,0) and (+.35,0) .. 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(-.35,1); \end{scope} \draw[line width=1.4] (1.05,1) to (1.05,-1); \begin{scope}[shift={(+2.4,0)}] \draw[line width=1.4] (-.5,5) to (-.5,-1); \draw[line width=1.2] (+.5,5) to (+.5,-1); \draw (-0,2.2) node {<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mpadded lspace='-50%width' width='0'><mi>⋯</mi></mpadded></mrow><annotation encoding='application/x-tex'>\mathclap{\cdots}</annotation></semantics></math>}; \end{scope} \draw (-2.1, -1.4) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math>}}; \draw (-1.1, -1.4) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mspace width='negativethinmathspace' /><mo>−</mo><mspace width='negativethinmathspace' /><mn>1</mn></mrow><annotation encoding='application/x-tex'>i\!-\!1</annotation></semantics></math>}}; \draw (-.4, -1.4) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi></mrow><annotation encoding='application/x-tex'>i</annotation></semantics></math>}}; \draw (+.35, -1.4) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mspace width='negativethinmathspace' /><mo>+</mo><mspace width='negativethinmathspace' /><mn>1</mn></mrow><annotation encoding='application/x-tex'>i\!+\!1</annotation></semantics></math>}}; \draw (+1.1, -1.4) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mspace width='negativethinmathspace' /><mo>+</mo><mspace width='negativethinmathspace' /><mn>2</mn></mrow><annotation encoding='application/x-tex'>i\!+\!2</annotation></semantics></math>}}; \draw (+1.9, -1.4) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mspace width='negativethinmathspace' /><mo>+</mo><mspace width='negativethinmathspace' /><mn>3</mn></mrow><annotation encoding='application/x-tex'>i\!+\!3</annotation></semantics></math>}}; \draw (+2.9, -1.4) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mspace width='negativethinmathspace' /><mo>+</mo><mspace width='negativethinmathspace' /><mn>1</mn></mrow><annotation encoding='application/x-tex'>n\!+\!1</annotation></semantics></math>}}; \end{tikzpicture} } \color{orange} \right] \color{black} <h1 id=''>\;\;\;\;\;</h1> \;\;\;\; \color{orange} \left[\color{black} \raisebox{-24pt}{ \begin{tikzpicture}yscale=.5] \begin{scope}[shift={(-1.6,0)}] \draw[line width=1.4] (-.5,5) to (-.5,-1); \draw[line width=1.2] (+.5,5) to (+.5,-1); \draw (-0,2.2) node {<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mpadded lspace='-50%width' width='0'><mi>⋯</mi></mpadded></mrow><annotation encoding='application/x-tex'>\mathclap{\cdots}</annotation></semantics></math>}; \end{scope} \begin{scope}[shift={(.7,4)}] \draw[line width=1.4] (-.35,-1) .. controls (-.35,0) and (+.35,0) .. (+.35,1); \draw[line width=4.5, white] (+.35,-1) .. controls (+.35,0) and (-.35,0) .. (-.35,1); \draw[line width=1.4] (+.35,-1) .. controls (+.35,0) and (-.35,0) .. (-.35,1); \end{scope} \draw[line width=1.4] <h2 id='1053'>(1.05,3)</h2> (1.05,1); \draw[line width=1.4] <h2 id='355'>(-.35,5)</h2> (-.35,3); \draw[line width=1.4] <h2 id='351_9'>(-.35,1)</h2> (-.35,-1); \begin{scope}[shift={(0,2)}] \draw[line width=1.4] (-.35,-1) .. controls (-.35,0) and (+.35,0) .. (+.35,1); \draw[line width=4.5, white] (+.35,-1) .. controls (+.35,0) and (-.35,0) .. (-.35,1); \draw[line width=1.4] (+.35,-1) .. controls (+.35,0) and (-.35,0) .. (-.35,1); \end{scope} \begin{scope}[shift={(.7,0)}] \draw[line width=1.4] (-.35,-1) .. controls (-.35,0) and (+.35,0) .. (+.35,1); \draw[line width=4.5, white] (+.35,-1) .. controls (+.35,0) and (-.35,0) .. (-.35,1); \draw[line width=1.4] (+.35,-1) .. controls (+.35,0) and (-.35,0) .. (-.35,1); \end{scope} \begin{scope}[shift={(+2.4,0)}] \draw[line width=1.4] (-.5,5) to (-.5,-1); \draw[line width=1.2] (+.5,5) to (+.5,-1); \draw (-0,2.2) node {<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mpadded lspace='-50%width' width='0'><mi>⋯</mi></mpadded></mrow><annotation encoding='application/x-tex'>\mathclap{\cdots}</annotation></semantics></math>}; \end{scope} \draw (-2.1, -1.4) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_72' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math>}}; \draw (-1.1, -1.4) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_73' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mspace width='negativethinmathspace' /><mo>−</mo><mspace width='negativethinmathspace' /><mn>1</mn></mrow><annotation encoding='application/x-tex'>i\!-\!1</annotation></semantics></math>}}; \draw (-.4, -1.4) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_74' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi></mrow><annotation encoding='application/x-tex'>i</annotation></semantics></math>}}; \draw (+.35, -1.4) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mspace width='negativethinmathspace' /><mo>+</mo><mspace width='negativethinmathspace' /><mn>1</mn></mrow><annotation encoding='application/x-tex'>i\!+\!1</annotation></semantics></math>}}; \draw (+1.1, -1.4) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_76' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mspace width='negativethinmathspace' /><mo>+</mo><mspace width='negativethinmathspace' /><mn>2</mn></mrow><annotation encoding='application/x-tex'>i\!+\!2</annotation></semantics></math>}}; \draw (+1.9, -1.4) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_77' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mspace width='negativethinmathspace' /><mo>+</mo><mspace width='negativethinmathspace' /><mn>3</mn></mrow><annotation encoding='application/x-tex'>i\!+\!3</annotation></semantics></math>}}; \draw (+2.9, -1.4) node {\color{blue}\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_78' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mspace width='negativethinmathspace' /><mo>+</mo><mspace width='negativethinmathspace' /><mn>1</mn></mrow><annotation encoding='application/x-tex'>n\!+\!1</annotation></semantics></math>}}; \end{tikzpicture} } \color{orange} \right] \color{black} \end{tikzcd} \end{definition} <h5 id='PresentationOfPureBraids'>Presentation of pure braids</h5> Similarly, the <a class='existingWikiWord' href='/nlab/show/diff/braid+group'>pure braid group</a> has a <a class='existingWikiWord' href='/nlab/show/diff/finitely+presentable+group'>finite presentation</a>. One possible set of generators (also originally considered by Artin) are the “weave” braids where one strands lassos exactly one other strand: <dl> <dt>\begin{tikzcd}</dt> <dt>b_{ij}</dt> <dt>\;\;</dt> <dd>= \;\; \color{orange} \left[\color{black} \raisebox{5pt}{ \begin{tikzpicture}yscale=1, xscale=1.3]</dd> </dl> \draw[line width=1.4] (.55,0) .. controls (.55, +.5) and (-.8, +.5) .. (-.8,+1); \begin{scope}[shift={(-1.4,0)}] \draw[line width=1.4] (-.3,1) to (-.3,-1); \draw[line width=1.4] (+.4,1) to (+.4,-1); \draw (+.05,-.1) node {<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_79' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mpadded lspace='-50%width' width='0'><mi>⋯</mi></mpadded></mrow><annotation encoding='application/x-tex'>\mathclap{\cdots}</annotation></semantics></math>}; \end{scope} \begin{scope}[shift={(-0,0)}] \draw[line width=1.4] (-.6,1) to (-.6,-1); \draw[line width=1.4] (-.4,1) to (-.4,-1); \draw[line width=4, white] (+.4,1) to (+.4,-1); \draw[line width=1.4] (+.4,1) to (+.4,-1); \draw[line width=4, white] (+.2,1) to (+.2,-1); \draw[line width=1.4] (+.2,1) to (+.2,-1); \draw (-.1,-.1) node {<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_80' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mpadded lspace='-50%width' width='0'><mi>⋯</mi></mpadded></mrow><annotation encoding='application/x-tex'>\mathclap{\cdots}</annotation></semantics></math>}; \end{scope} \begin{scope}[shift={(+.8,0)}] \draw[line width=1.4] (-.4,1) to (-.4,-1); \draw[line width=1.2] (+.3,1) to (+.3,-1); \draw (-.01,-.1) node {<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_81' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mpadded lspace='-50%width' width='0'><mi>⋯</mi></mpadded></mrow><annotation encoding='application/x-tex'>\mathclap{\cdots}</annotation></semantics></math>}; \end{scope} \draw[line width=4, white] (-.8,-1) .. controls (-.8, -.5) and (.55, -.5) .. (.55,0); \draw[line width=1.4] (-.8,-1) .. controls (-.8, -.5) and (.55, -.5) .. (.55,0); \begin{scope} \clip (-.9,0) rectangle (+.3,1.2); \draw[line width=4, white] (.55,0) .. controls (.55, +.5) and (-.8, +.5) .. (-.8,+1); \draw[line width=1.4] (.55,0) .. controls (.55, +.5) and (-.8, +.5) .. (-.8,+1); \end{scope} \draw (-.8,-1.3) node { \scalebox{ \color{blue} <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_82' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi></mrow><annotation encoding='application/x-tex'>i</annotation></semantics></math> } }; \draw (+.4,-1.3) node { \scalebox{ \color{blue} <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_83' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>j</mi></mrow><annotation encoding='application/x-tex'>j</annotation></semantics></math> } }; \end{tikzpicture} } \color{orange} \right] \color{black} <h1 id='_2'>\;\;</h1> \;\; \color{orange} \left[\color{black} \raisebox{5pt}{ \begin{tikzpicture}yscale=-1, xscale=-1.3] \draw[line width=1.4] (.55,0) .. controls (.55, +.5) and (-.8, +.5) .. (-.8,+1); \begin{scope}[shift={(-1.4,0)}] \draw[line width=1.4] (-.3,1) to (-.3,-1); \draw[line width=1.4] (+.4,1) to (+.4,-1); \draw (+.05,-.1) node {<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_84' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mpadded lspace='-50%width' width='0'><mi>⋯</mi></mpadded></mrow><annotation encoding='application/x-tex'>\mathclap{\cdots}</annotation></semantics></math>}; \end{scope} \begin{scope}[shift={(-0,0)}] \draw[line width=1.4] (-.6,1) to (-.6,-1); \draw[line width=1.4] (-.4,1) to (-.4,-1); \draw[line width=4, white] (+.4,1) to (+.4,-1); \draw[line width=1.4] (+.4,1) to (+.4,-1); \draw[line width=4, white] (+.2,1) to (+.2,-1); \draw[line width=1.4] (+.2,1) to (+.2,-1); \draw (-.1,-.1) node {<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_85' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mpadded lspace='-50%width' width='0'><mi>⋯</mi></mpadded></mrow><annotation encoding='application/x-tex'>\mathclap{\cdots}</annotation></semantics></math>}; \end{scope} \begin{scope}[shift={(+.8,0)}] \draw[line width=1.4] (-.4,1) to (-.4,-1); \draw[line width=1.2] (+.3,1) to (+.3,-1); \draw (-.01,-.1) node {<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_86' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mpadded lspace='-50%width' width='0'><mi>⋯</mi></mpadded></mrow><annotation encoding='application/x-tex'>\mathclap{\cdots}</annotation></semantics></math>}; \end{scope} \draw[line width=4, white] (-.8,-1) .. controls (-.8, -.5) and (.55, -.5) .. (.55,0); \draw[line width=1.4] (-.8,-1) .. controls (-.8, -.5) and (.55, -.5) .. (.55,0); \begin{scope} \clip (-.9,0) rectangle (+.3,1.2); \draw[line width=4, white] (.55,0) .. controls (.55, +.5) and (-.8, +.5) .. (-.8,+1); \draw[line width=1.4] (.55,0) .. controls (.55, +.5) and (-.8, +.5) .. (-.8,+1); \end{scope} \draw (-.8,+1.3) node { \scalebox{-.7}{ \color{blue} <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_87' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>j</mi></mrow><annotation encoding='application/x-tex'>j</annotation></semantics></math> } }; \draw (+.4,+1.3) node { \scalebox{-.7}{ \color{blue} <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_88' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi></mrow><annotation encoding='application/x-tex'>i</annotation></semantics></math> } }; \end{tikzpicture} } \color{orange} \right] \color{black} \end{tikzcd} In terms of these generators, the <a class='existingWikiWord' href='/nlab/show/diff/braid+group'>pure braid group</a> is obained by quotienting out the following relations — an optimization of Artin’s original pure braid relations, due to <a href='#Lee10'>Lee (2010, Thm. 1.1, Rem. 3.1)</a>: <img src='/nlab/files/PureBraidGroupPresentation-230205b.jpg' width='830' /> \begin{tikzcd} \color{orange} \left[\color{black} \raisebox{7pt}{ \begin{tikzpicture}line width=1.2] \begin{scope}[shift={(0,+1)}] \begin{scope}[shift={(-2,0)}, gray] \draw (-.75,-1) to (-.75,1); \draw (-.25,-1) to (-.25,1); \draw (-.5,-.6) node {\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_89' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi></mrow><annotation encoding='application/x-tex'>\cdots</annotation></semantics></math>}}; \end{scope} \draw (-2,-1) to (-2,1); \begin{scope}[shift={(-1,0)}, gray] \draw (-.75,-1) to (-.75,1); \draw (-.25,-1) to (-.25,1); \draw (-.5,-.6) node {\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_90' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi></mrow><annotation encoding='application/x-tex'>\cdots</annotation></semantics></math>}}; \end{scope} \draw (-1,-1) to (-1,1); \begin{scope}[shift={(0,0)}, gray] \draw (-.75,-1) to (-.75,1); \draw (-.25,-1) to (-.25,1); \draw (-.5,-.6) node {\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_91' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi></mrow><annotation encoding='application/x-tex'>\cdots</annotation></semantics></math>}}; \end{scope} \begin{scope}[shift={(+1,0)}, gray] \draw (-.75,-1) to (-.75,1); \draw (-.25,-1) to (-.25,1); \draw (-.5,-.6) node {\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_92' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi></mrow><annotation encoding='application/x-tex'>\cdots</annotation></semantics></math>}}; \end{scope} \draw (+1,-1) to (+1,1); \begin{scope}[shift={(+2,0)}, gray] \draw (-.75,-1) to (-.75,1); \draw (-.25,-1) to (-.25,1); \draw (-.5,-.6) node {\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_93' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi></mrow><annotation encoding='application/x-tex'>\cdots</annotation></semantics></math>}}; \end{scope} \begin{scope}[shift={(2,0)}] \draw[line width=4, white] <h2 id='21'>(-2,-1)</h2> (-2,-.8) .. controls (-2,-.4) and (-.85, -.4) .. 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(-1.15, 0) .. controls (-1.15, .4) and (+1, .4) .. <h2 id='1_8_16'>(+1, .8)</h2> (+1, 1); \draw[line width=4, white] (-1,-1) to (-1,0); \draw (-1,-1) to (-1,0); \end{scope} \begin{scope}[shift={(0,+1)}] \begin{scope}[shift={(-1,0)}, gray] \draw (-.75,-1) to (-.75,1); \draw (-.25,-1) to (-.25,1); \draw (-.5,-.6) node {\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_202' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi></mrow><annotation encoding='application/x-tex'>\cdots</annotation></semantics></math>}}; \end{scope} \draw (-1,-1) to (-1,1); \begin{scope}[shift={(0,0)}, gray] \draw (-.75,-1) to (-.75,1); \draw (-.25,-1) to (-.25,1); \draw (-.5,-.6) node {\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_203' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi></mrow><annotation encoding='application/x-tex'>\cdots</annotation></semantics></math>}}; \end{scope} \draw (0,-1) to (0,1); \begin{scope}[shift={(+1,0)}, gray] \draw (-.75,-1) to (-.75,1); \draw (-.25,-1) to (-.25,1); \draw (-.5,-.6) node {\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_204' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi></mrow><annotation encoding='application/x-tex'>\cdots</annotation></semantics></math>}}; \end{scope} \draw (+1,-1) to (+1,1); \begin{scope}[shift={(+2,0)}, gray] \draw (-.75,-1) to (-.75,1); \draw (-.25,-1) to (-.25,1); \draw (-.5,-.6) node {\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_205' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi></mrow><annotation encoding='application/x-tex'>\cdots</annotation></semantics></math>}}; \end{scope} \begin{scope}[shift={(+3,0)}, gray] \draw (-.75,-1) to (-.75,1); \draw (-.25,-1) to (-.25,1); \draw (-.5,-.6) node {\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_206' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi></mrow><annotation encoding='application/x-tex'>\cdots</annotation></semantics></math>}}; \end{scope} \draw[line width=4, white] <h2 id='2_1_3'>(+2, -1)</h2> (+2, -.8) .. controls (2, -.4) and (-1.15, -.4) .. (-1.15, 0) .. controls (-1.15, +.4) and (2, +.4) .. <h2 id='2_8_3'>(+2, +.8)</h2> (+2, 1); \draw <h2 id='2_1_4'>(+2, -1)</h2> (+2, -.8) .. controls (2, -.4) and (-1.15, -.4) .. (-1.15, 0) .. controls (-1.15, +.4) and (2, +.4) .. <h2 id='2_8_4'>(+2, +.8)</h2> (+2, 1); \draw[line width=4, white] (-1,-1) to (-1,0); \draw (-1,-1) to (-1,0); \draw[line width=4, white] (0,-1) to (0,0); \draw (0,-1) to (0,0); \draw[line width=4, white] (1,-1) to (1,0); \draw (1,-1) to (1,0); \end{scope} \node at (-1, -2.4) { \color{blue} <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_207' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>r</mi></mrow><annotation encoding='application/x-tex'>r</annotation></semantics></math> }; \node at (0, -2.4) { \color{blue} <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_208' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi></mrow><annotation encoding='application/x-tex'>i</annotation></semantics></math> }; \node at (+1, -2.4) { \color{blue} <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_209' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>s</mi></mrow><annotation encoding='application/x-tex'>s</annotation></semantics></math> }; \node at (+2, -2.4) { \color{blue} <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_210' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>j</mi></mrow><annotation encoding='application/x-tex'>j</annotation></semantics></math> }; \end{tikzpicture} } \color{orange} \right] \color{black} \end{tikzcd} Here we repeatedly used products of pure braid generators of the following form: \begin{tikzcd} \color{orange} \left[\color{black} \raisebox{30pt}{ \begin{tikzpicture}line width=1.2] \begin{scope}[shift={(0,-3)}] \begin{scope}[shift={(-1,0)}, gray] \draw (-.75,-1) to (-.75,1); \draw (-.25,-1) to (-.25,1); \draw (-.5,-.6) node {\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_211' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi></mrow><annotation encoding='application/x-tex'>\cdots</annotation></semantics></math>}}; \end{scope} \begin{scope}[shift={(0,0)}, gray] \draw (-.75,-1) to (-.75,1); \draw (-.25,-1) to (-.25,1); \draw (-.5,-.6) node {\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_212' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi></mrow><annotation encoding='application/x-tex'>\cdots</annotation></semantics></math>}}; \end{scope} \draw (0,-1) to (0,1); \begin{scope}[shift={(+1,0)}, gray] \draw (-.75,-1) to (-.75,1); \draw (-.25,-1) to (-.25,1); \draw (-.5,-.6) node {\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_213' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi></mrow><annotation encoding='application/x-tex'>\cdots</annotation></semantics></math>}}; \end{scope} \draw (+1,-1) to (+1,1); \begin{scope}[shift={(+2,0)}, gray] \draw (-.75,-1) to (-.75,1); \draw (-.25,-1) to (-.25,1); \draw (-.5,-.6) node {\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_214' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi></mrow><annotation encoding='application/x-tex'>\cdots</annotation></semantics></math>}}; \end{scope} \draw (+2,-1) to (+2,1); \begin{scope}[shift={(+3,0)}, gray] \draw (-.75,-1) to (-.75,1); \draw (-.25,-1) to (-.25,1); \draw (-.5,-.6) node {\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_215' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi></mrow><annotation encoding='application/x-tex'>\cdots</annotation></semantics></math>}}; \end{scope} \draw[line width=4pt, white] <h2 id='11_17'>(-1,-1)</h2> (-1, -.8) .. controls (-1, -.4) and (2.15, -.4) .. (2.15, 0) .. controls (2.15, .4) and (-1, .4) .. <h2 id='1_8_17'>(-1, .8)</h2> (-1,1); \draw <h2 id='11_18'>(-1,-1)</h2> (-1, -.8) .. controls (-1, -.4) and (2.15, -.4) .. (2.15, 0) .. controls (2.15, .4) and (-1, .4) .. <h2 id='1_8_18'>(-1, .8)</h2> (-1,1); \draw[line width=4pt, white] (2, 0) to (2,1); \draw (2, 0) to (2,1); \end{scope} \begin{scope}[shift={(0,-1)}] \begin{scope}[shift={(-1,0)}, gray] \draw (-.75,-1) to (-.75,1); \draw (-.25,-1) to (-.25,1); \draw (-.5,-.6) node {\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_216' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi></mrow><annotation encoding='application/x-tex'>\cdots</annotation></semantics></math>}}; \end{scope} \begin{scope}[shift={(0,0)}, gray] \draw (-.75,-1) to (-.75,1); \draw (-.25,-1) to (-.25,1); \draw (-.5,-.6) node {\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_217' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi></mrow><annotation encoding='application/x-tex'>\cdots</annotation></semantics></math>}}; \end{scope} \draw (0,-1) to (0,1); \begin{scope}[shift={(+1,0)}, gray] \draw (-.75,-1) to (-.75,1); \draw (-.25,-1) to (-.25,1); \draw (-.5,-.6) node {\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_218' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi></mrow><annotation encoding='application/x-tex'>\cdots</annotation></semantics></math>}}; \end{scope} \draw (+1,-1) to (+1,1); \begin{scope}[shift={(+2,0)}, gray] \draw (-.75,-1) to (-.75,1); \draw (-.25,-1) to (-.25,1); \draw (-.5,-.6) node {\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_219' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi></mrow><annotation encoding='application/x-tex'>\cdots</annotation></semantics></math>}}; \end{scope} \draw (+2,-1) to (+2,1); \begin{scope}[shift={(+3,0)}, gray] \draw (-.75,-1) to (-.75,1); \draw (-.25,-1) to (-.25,1); \draw (-.5,-.6) node {\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_220' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi></mrow><annotation encoding='application/x-tex'>\cdots</annotation></semantics></math>}}; \end{scope} \draw[line width=4pt, white] <h2 id='11_19'>(-1,-1)</h2> (-1, -.8) .. controls (-1, -.4) and (1.15, -.4) .. (1.15, 0) .. controls (1.15, .4) and (-1, .4) .. <h2 id='1_8_19'>(-1, .8)</h2> (-1,1); \draw <h2 id='11_20'>(-1,-1)</h2> (-1, -.8) .. controls (-1, -.4) and (1.15, -.4) .. (1.15, 0) .. controls (1.15, .4) and (-1, .4) .. <h2 id='1_8_20'>(-1, .8)</h2> (-1,1); \draw[line width=4pt, white] (1, 0) to (1,1); \draw (1, 0) to (1,1); \end{scope} \begin{scope}[shift={(0,+1)}] \begin{scope}[shift={(-1,0)}, gray] \draw (-.75,-1) to (-.75,1); \draw (-.25,-1) to (-.25,1); \draw (-.5,-.6) node {\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_221' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi></mrow><annotation encoding='application/x-tex'>\cdots</annotation></semantics></math>}}; \end{scope} \begin{scope}[shift={(0,0)}, gray] \draw (-.75,-1) to (-.75,1); \draw (-.25,-1) to (-.25,1); \draw (-.5,-.6) node {\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_222' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi></mrow><annotation encoding='application/x-tex'>\cdots</annotation></semantics></math>}}; \end{scope} \draw (0,-1) to (0,1); \begin{scope}[shift={(+1,0)}, gray] \draw (-.75,-1) to (-.75,1); \draw (-.25,-1) to (-.25,1); \draw (-.5,-.6) node {\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_223' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi></mrow><annotation encoding='application/x-tex'>\cdots</annotation></semantics></math>}}; \end{scope} \draw (+1,-1) to (+1,1); \begin{scope}[shift={(+2,0)}, gray] \draw (-.75,-1) to (-.75,1); \draw (-.25,-1) to (-.25,1); \draw (-.5,-.6) node {\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_224' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi></mrow><annotation encoding='application/x-tex'>\cdots</annotation></semantics></math>}}; \end{scope} \draw (+2,-1) to (+2,1); \begin{scope}[shift={(+3,0)}, gray] \draw (-.75,-1) to (-.75,1); \draw (-.25,-1) to (-.25,1); \draw (-.5,-.6) node {\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_225' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi></mrow><annotation encoding='application/x-tex'>\cdots</annotation></semantics></math>}}; \end{scope} \draw[line width=4, white] <h2 id='11_21'>(-1,-1)</h2> (-1,-.8) .. controls (-1, -.4) and (.15, -.4) .. (.15, 0) .. controls (.15, .4) and (-1, .4) .. <h2 id='1_8_21'>(-1, .8)</h2> (-1, 1); \draw <h2 id='11_22'>(-1,-1)</h2> (-1,-.8) .. controls (-1, -.4) and (.15, -.4) .. (.15, 0) .. controls (.15, .4) and (-1, .4) .. <h2 id='1_8_22'>(-1, .8)</h2> (-1, 1); \draw[line width=4, white] (0, 0) to (0,1); \draw (0, 0) to (0,1); \end{scope} \end{tikzpicture} } \color{orange} \right] \color{black} <h1 id='_8'>\;\;\;</h1> \;\;\; \color{orange} \left[\color{black} \raisebox{30pt}{ \begin{tikzpicture}line width=1.2] \begin{scope}[shift={(0,-2)}, yscale=2] \begin{scope}[shift={(-1,0)}, gray] \draw (-.75,-1) to (-.75,1); \draw (-.25,-1) to (-.25,1); \draw (-.5,-.6) node {\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_226' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi></mrow><annotation encoding='application/x-tex'>\cdots</annotation></semantics></math>}}; \end{scope} \begin{scope}[shift={(0,0)}, gray] \draw (-.75,-1) to (-.75,1); \draw (-.25,-1) to (-.25,1); \draw (-.5,-.6) node {\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_227' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi></mrow><annotation encoding='application/x-tex'>\cdots</annotation></semantics></math>}}; \end{scope} \draw (0,-1) to (0,1); \begin{scope}[shift={(+1,0)}, gray] \draw (-.75,-1) to (-.75,1); \draw (-.25,-1) to (-.25,1); \draw (-.5,-.6) node {\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_228' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi></mrow><annotation encoding='application/x-tex'>\cdots</annotation></semantics></math>}}; \end{scope} \draw (+1,-1) to (+1,1); \begin{scope}[shift={(+2,0)}, gray] \draw (-.75,-1) to (-.75,1); \draw (-.25,-1) to (-.25,1); \draw (-.5,-.6) node {\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_229' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi></mrow><annotation encoding='application/x-tex'>\cdots</annotation></semantics></math>}}; \end{scope} \draw (+2,-1) to (+2,1); \begin{scope}[shift={(+3,0)}, gray] \draw (-.75,-1) to (-.75,1); \draw (-.25,-1) to (-.25,1); \draw (-.5,-.6) node {\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_230' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi></mrow><annotation encoding='application/x-tex'>\cdots</annotation></semantics></math>}}; \end{scope} \draw[line width=4pt, white] <h2 id='11_23'>(-1,-1)</h2> (-1, -.8) .. controls (-1, -.4) and (2.15, -.4) .. (2.15, 0) .. controls (2.15, .4) and (-1, .4) .. <h2 id='1_8_23'>(-1, .8)</h2> (-1,1); \draw <h2 id='11_24'>(-1,-1)</h2> (-1, -.8) .. controls (-1, -.4) and (2.15, -.4) .. (2.15, 0) .. controls (2.15, .4) and (-1, .4) .. <h2 id='1_8_24'>(-1, .8)</h2> (-1,1); \draw[line width=4pt, white] (2, 0) to (2,1); \draw (2, 0) to (2,1); \draw[line width=4pt, white] (1, 0) to (1,1); \draw (1, 0) to (1,1); \end{scope} \begin{scope}[shift={(0,+1)}] \begin{scope}[shift={(-1,0)}, gray] \draw (-.75,-1) to (-.75,1); \draw (-.25,-1) to (-.25,1); \draw (-.5,-.6) node {\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_231' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi></mrow><annotation encoding='application/x-tex'>\cdots</annotation></semantics></math>}}; \end{scope} \begin{scope}[shift={(0,0)}, gray] \draw (-.75,-1) to (-.75,1); \draw (-.25,-1) to (-.25,1); \draw (-.5,-.6) node {\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_232' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi></mrow><annotation encoding='application/x-tex'>\cdots</annotation></semantics></math>}}; \end{scope} \draw (0,-1) to (0,1); \begin{scope}[shift={(+1,0)}, gray] \draw (-.75,-1) to (-.75,1); \draw (-.25,-1) to (-.25,1); \draw (-.5,-.6) node {\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_233' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi></mrow><annotation encoding='application/x-tex'>\cdots</annotation></semantics></math>}}; \end{scope} \draw (+1,-1) to (+1,1); \begin{scope}[shift={(+2,0)}, gray] \draw (-.75,-1) to (-.75,1); \draw (-.25,-1) to (-.25,1); \draw (-.5,-.6) node {\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_234' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi></mrow><annotation encoding='application/x-tex'>\cdots</annotation></semantics></math>}}; \end{scope} \draw (+2,-1) to (+2,1); \begin{scope}[shift={(+3,0)}, gray] \draw (-.75,-1) to (-.75,1); \draw (-.25,-1) to (-.25,1); \draw (-.5,-.6) node {\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_235' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi></mrow><annotation encoding='application/x-tex'>\cdots</annotation></semantics></math>}}; \end{scope} \draw[line width=4, white] <h2 id='11_25'>(-1,-1)</h2> (-1,-.8) .. controls (-1, -.4) and (.15, -.4) .. (.15, 0) .. controls (.15, .4) and (-1, .4) .. <h2 id='1_8_25'>(-1, .8)</h2> (-1, 1); \draw <h2 id='11_26'>(-1,-1)</h2> (-1,-.8) .. controls (-1, -.4) and (.15, -.4) .. (.15, 0) .. controls (.15, .4) and (-1, .4) .. <h2 id='1_8_26'>(-1, .8)</h2> (-1, 1); \draw[line width=4, white] (0, 0) to (0,1); \draw (0, 0) to (0,1); \end{scope} \end{tikzpicture} } \color{orange} \right] \color{black} <h1 id='_9'>\;\;\;</h1> \;\;\; \color{orange} \left[\color{black} \raisebox{30pt}{ \begin{tikzpicture}line width=1.2] \begin{scope}[shift={(0,-1)}, yscale=3] \begin{scope}[shift={(-1,0)}, gray] \draw (-.75,-1) to (-.75,1); \draw (-.25,-1) to (-.25,1); \draw (-.5,-.6) node {\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_236' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi></mrow><annotation encoding='application/x-tex'>\cdots</annotation></semantics></math>}}; \end{scope} \begin{scope}[shift={(0,0)}, gray] \draw (-.75,-1) to (-.75,1); \draw (-.25,-1) to (-.25,1); \draw (-.5,-.6) node {\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_237' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi></mrow><annotation encoding='application/x-tex'>\cdots</annotation></semantics></math>}}; \end{scope} \draw (0,-1) to (0,1); \begin{scope}[shift={(+1,0)}, gray] \draw (-.75,-1) to (-.75,1); \draw (-.25,-1) to (-.25,1); \draw (-.5,-.6) node {\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_238' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi></mrow><annotation encoding='application/x-tex'>\cdots</annotation></semantics></math>}}; \end{scope} \draw (+1,-1) to (+1,1); \begin{scope}[shift={(+2,0)}, gray] \draw (-.75,-1) to (-.75,1); \draw (-.25,-1) to (-.25,1); \draw (-.5,-.6) node {\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_239' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi></mrow><annotation encoding='application/x-tex'>\cdots</annotation></semantics></math>}}; \end{scope} \draw (+2,-1) to (+2,1); \begin{scope}[shift={(+3,0)}, gray] \draw (-.75,-1) to (-.75,1); \draw (-.25,-1) to (-.25,1); \draw (-.5,-.6) node {\scalebox{<math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_240' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi></mrow><annotation encoding='application/x-tex'>\cdots</annotation></semantics></math>}}; \end{scope} \draw[line width=4pt, white] <h2 id='11_27'>(-1,-1)</h2> (-1, -.8) .. controls (-1, -.4) and (2.15, -.4) .. (2.15, 0) .. controls (2.15, .4) and (-1, .4) .. <h2 id='1_8_27'>(-1, .8)</h2> (-1,1); \draw <h2 id='11_28'>(-1,-1)</h2> (-1, -.8) .. controls (-1, -.4) and (2.15, -.4) .. (2.15, 0) .. controls (2.15, .4) and (-1, .4) .. <h2 id='1_8_28'>(-1, .8)</h2> (-1,1); \draw[line width=4pt, white] (2, 0) to (2,1); \draw (2, 0) to (2,1); \draw[line width=4pt, white] (1, 0) to (1,1); \draw (1, 0) to (1,1); \draw[line width=4pt, white] (0, 0) to (0,1); \draw (0, 0) to (0,1); \end{scope} \end{tikzpicture} } \color{orange} \right] \color{black} \end{tikzcd} <h4 id='AsFundamentalGroupOfConfigurationSpace'>As fundamental group of a configuration space of points</h4> Geometrically, one may understand the group of braids in <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_241' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝ</mi> <mn>3</mn></msup></mrow><annotation encoding='application/x-tex'>\mathbb{R}^3</annotation></semantics></math> as the <a class='existingWikiWord' href='/nlab/show/diff/fundamental+group'>fundamental group</a> of the <a class='existingWikiWord' href='/nlab/show/diff/configuration+space+of+points'>configuration space of points</a> in the <a class='existingWikiWord' href='/nlab/show/diff/plane'>plane</a> <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_242' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding='application/x-tex'>\mathbb{R}^2</annotation></semantics></math> (traditionally regarded as the <a class='existingWikiWord' href='/nlab/show/diff/complex+number'>complex plane</a> <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_243' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding='application/x-tex'>\mathbb{C}</annotation></semantics></math> in this context, though the <a class='existingWikiWord' href='/nlab/show/diff/complex+structure'>complex structure</a> plays no role in the definition of the braid group as such). (originally due to <a href='#Hurwitz1891'>Hurwitz 1891, §II</a>, then re-discovered/re-vived in <a href='#FadellNeuwirth62'>Fadell & Neuwirth 1962, p. 118</a>, <a href='#FoxNeuwirth62'>Fox & Neuwirth 1962, §7</a>, review includes <a href='#Birman75'>Birman 1975, §1</a>, <a href='#Williams20'>Williams 2020, pp. 9</a>) \linebreak In the plane We say this in more detail: Let <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_244' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>C</mi> <mi>n</mi></msub><mo>↪</mo><msup><mi>ℂ</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>C_n \hookrightarrow \mathbb{C}^n</annotation></semantics></math> denote the space of <a class='existingWikiWord' href='/nlab/show/diff/configuration+space+of+points'>configurations of n ordered points</a> in the <a class='existingWikiWord' href='/nlab/show/diff/complex+number'>complex plane</a>, whose elements are those <a class='existingWikiWord' href='/nlab/show/diff/tuple'>n-tuples</a> <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_245' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>z</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>z</mi> <mi>n</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(z_1, \ldots, z_n)</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_246' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>z</mi> <mi>i</mi></msub><mo>≠</mo><msub><mi>z</mi> <mi>j</mi></msub></mrow><annotation encoding='application/x-tex'>z_i \neq z_j</annotation></semantics></math> whenever <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_247' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo>≠</mo><mi>j</mi></mrow><annotation encoding='application/x-tex'>i \neq j</annotation></semantics></math>. In other words, <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_248' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>C</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>C_n</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/complement'>complement</a> of the <a class='existingWikiWord' href='/nlab/show/diff/fat+diagonal'>fat diagonal</a>: <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_249' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>C</mi> <mi>n</mi></msub><mspace width='thickmathspace' /><mo>≔</mo><mspace width='thickmathspace' /><msup><mi>ℂ</mi> <mi>n</mi></msup><mo>∖</mo><msubsup><mstyle mathvariant='bold'><mi>Δ</mi></mstyle> <mi>ℂ</mi> <mi>n</mi></msubsup><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> C_n \;\coloneqq\; \mathbb{C}^n \setminus \mathbf{\Delta}^n_{\mathbb{C}} \,. </annotation></semantics></math></div> The <a class='existingWikiWord' href='/nlab/show/diff/symmetric+group'>symmetric group</a> <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_250' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>S</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>S_n</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/action'>acts</a> on <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_251' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>C</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>C_n</annotation></semantics></math> by <a class='existingWikiWord' href='/nlab/show/diff/permutation'>permuting</a> coordinates. Let: <ul> <li> <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_252' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>C</mi> <mi>n</mi></msub><mo stretchy='false'>/</mo><msub><mi>S</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>C_n/S_n</annotation></semantics></math> denote the <a class='existingWikiWord' href='/nlab/show/diff/quotient+object'>quotient</a> by this group action, hence the <a class='existingWikiWord' href='/nlab/show/diff/orbit'>orbit</a> space (the space of <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_253' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-element subsets of <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_254' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding='application/x-tex'>\mathbb{C}</annotation></semantics></math> if one likes), </li> <li> <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_255' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msub><mi>z</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>z</mi> <mi>n</mi></msub><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[z_1, \ldots, z_n]</annotation></semantics></math> denote the image of <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_256' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>z</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>z</mi> <mi>n</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(z_1, \ldots, z_n)</annotation></semantics></math> under the quotient <a class='existingWikiWord' href='/nlab/show/diff/coprojection'>coprojection</a> <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_257' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>π</mi><mo lspace='verythinmathspace'>:</mo><msub><mi>C</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>C</mi> <mi>n</mi></msub><mo stretchy='false'>/</mo><msub><mi>S</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>\pi \colon C_n \to C_n/S_n</annotation></semantics></math> (i.e. its the <a class='existingWikiWord' href='/nlab/show/diff/equivalence+class'>equivalence class</a>). </li> </ul> We understand <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_258' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo>=</mo><mo stretchy='false'>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi>…</mi><mo>,</mo><mi>n</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>p = (1, 2, \ldots, n)</annotation></semantics></math> as the basepoint for <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_259' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>C</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>C_n</annotation></semantics></math>, and <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_260' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>p</mi><mo stretchy='false'>]</mo><mo>=</mo><mo stretchy='false'>[</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi>…</mi><mi>n</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[p] = [1, 2, \ldots n]</annotation></semantics></math> as the basepoint for the <a class='existingWikiWord' href='/nlab/show/diff/configuration+space+of+points'>configuration space of unordered points</a> <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_261' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>C</mi> <mi>n</mi></msub><mo stretchy='false'>/</mo><msub><mi>S</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>C_n/S_n</annotation></semantics></math>, making it a <a class='existingWikiWord' href='/nlab/show/diff/pointed+topological+space'>pointed topological space</a>. <div class='num_defn'> <h6 id='definition'>Definition</h6> The braid group is the <a class='existingWikiWord' href='/nlab/show/diff/fundamental+group'>fundamental group</a> of the <a class='existingWikiWord' href='/nlab/show/diff/configuration+space+of+points'>configuration space of n unordered points</a>: <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_262' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Br</mi><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo>≔</mo><mspace width='thickmathspace' /><msub><mi>π</mi> <mn>1</mn></msub><mo maxsize='1.2em' minsize='1.2em'>(</mo><msub><mi>C</mi> <mi>n</mi></msub><mo stretchy='false'>/</mo><msub><mi>S</mi> <mi>n</mi></msub><mo>,</mo><mo stretchy='false'>[</mo><mi>p</mi><mo stretchy='false'>]</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo></mrow><annotation encoding='application/x-tex'> Br(n) \;\coloneqq\; \pi_1 \big( C_n/S_n, [p] \big) </annotation></semantics></math></div> The pure braid group is the <a class='existingWikiWord' href='/nlab/show/diff/fundamental+group'>fundamental group</a> of the <a class='existingWikiWord' href='/nlab/show/diff/configuration+space+of+points'>configuration space of n ordered points</a>: <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_263' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>PBr</mi><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo>≔</mo><mspace width='thickmathspace' /><msub><mi>π</mi> <mn>1</mn></msub><mo maxsize='1.2em' minsize='1.2em'>(</mo><msub><mi>C</mi> <mi>n</mi></msub><mo>,</mo><mi>p</mi><mo maxsize='1.2em' minsize='1.2em'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> PBr(n) \;\coloneqq\; \pi_1 \big( C_n, p \big) \,. </annotation></semantics></math></div></div> Evidently a braid <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_264' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>β</mi></mrow><annotation encoding='application/x-tex'>\beta</annotation></semantics></math> is represented by a path <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_265' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>α</mi><mo>:</mo><mi>I</mi><mo>→</mo><msub><mi>C</mi> <mi>n</mi></msub><mo stretchy='false'>/</mo><msub><mi>S</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>\alpha: I \to C_n/S_n</annotation></semantics></math> with <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_266' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>α</mi><mo stretchy='false'>(</mo><mn>0</mn><mo stretchy='false'>)</mo><mo>=</mo><mo stretchy='false'>[</mo><mi>p</mi><mo stretchy='false'>]</mo><mo>=</mo><mi>α</mi><mo stretchy='false'>(</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\alpha(0) = [p] = \alpha(1)</annotation></semantics></math>. Such a path may be uniquely lifted through the <a class='existingWikiWord' href='/nlab/show/diff/covering'>covering</a> projection <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_267' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>π</mi><mo>:</mo><msub><mi>C</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>C</mi> <mi>n</mi></msub><mo stretchy='false'>/</mo><msub><mi>S</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>\pi: C_n \to C_n/S_n</annotation></semantics></math> to a path <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_268' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>α</mi><mo stretchy='false'>˜</mo></mover></mrow><annotation encoding='application/x-tex'>\tilde{\alpha}</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_269' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>α</mi><mo stretchy='false'>˜</mo></mover><mo stretchy='false'>(</mo><mn>0</mn><mo stretchy='false'>)</mo><mo>=</mo><mi>p</mi></mrow><annotation encoding='application/x-tex'>\tilde{\alpha}(0) = p</annotation></semantics></math>. The end of the path <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_270' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>α</mi><mo stretchy='false'>˜</mo></mover><mo stretchy='false'>(</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\tilde{\alpha}(1)</annotation></semantics></math> has the same underlying subset as <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_271' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math> but with coordinates permuted: <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_272' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>α</mi><mo stretchy='false'>˜</mo></mover><mo stretchy='false'>(</mo><mn>1</mn><mo stretchy='false'>)</mo><mo>=</mo><mo stretchy='false'>(</mo><mi>σ</mi><mo stretchy='false'>(</mo><mn>1</mn><mo stretchy='false'>)</mo><mo>,</mo><mi>σ</mi><mo stretchy='false'>(</mo><mn>2</mn><mo stretchy='false'>)</mo><mo>,</mo><mi>…</mi><mo>,</mo><mi>σ</mi><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\tilde{\alpha}(1) = (\sigma(1), \sigma(2), \ldots, \sigma(n))</annotation></semantics></math>. Thus the braid <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_273' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>β</mi></mrow><annotation encoding='application/x-tex'>\beta</annotation></semantics></math> is exhibited by <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_274' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> non-intersecting strands, each one connecting an <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_275' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi></mrow><annotation encoding='application/x-tex'>i</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_276' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>σ</mi><mo stretchy='false'>(</mo><mi>i</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\sigma(i)</annotation></semantics></math>, and we have a map <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_277' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>β</mi><mo>↦</mo><mi>σ</mi></mrow><annotation encoding='application/x-tex'>\beta \mapsto \sigma</annotation></semantics></math> appearing as the quotient map of an exact sequence <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_278' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn><mo>→</mo><mi>PBr</mi><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>Br</mi><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>Sym</mi><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo><mo>→</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>1 \to PBr(n) \to Br(n) \to Sym(n) \to 1</annotation></semantics></math></div> which is part of a <a class='existingWikiWord' href='/nlab/show/diff/long+exact+sequence+of+homotopy+groups'>long exact homotopy sequence</a> corresponding to the <a class='existingWikiWord' href='/nlab/show/diff/fibration'>fibration</a> <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_279' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>π</mi><mo lspace='verythinmathspace'>:</mo><msub><mi>C</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>C</mi> <mi>n</mi></msub><mo stretchy='false'>/</mo><msub><mi>S</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>\pi \colon C_n \to C_n/S_n</annotation></semantics></math>. \linebreak In general surfaces or graphs Since the notion of a <a class='existingWikiWord' href='/nlab/show/diff/configuration+space+of+points'>configuration space of points</a> makes sense for points in any <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a>, not necessarily the <a class='existingWikiWord' href='/nlab/show/diff/plane'>plane</a> <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_280' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding='application/x-tex'>\mathbb{R}^2</annotation></semantics></math>, the <a href='#GeometricDefinition'>above</a> geometric definition has an immediate generalization: For <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_281' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Σ</mi></mrow><annotation encoding='application/x-tex'>\Sigma</annotation></semantics></math> any <a class='existingWikiWord' href='/nlab/show/diff/surface'>surface</a>, the <a class='existingWikiWord' href='/nlab/show/diff/fundamental+group'>fundamental group</a> of the (ordered) <a class='existingWikiWord' href='/nlab/show/diff/configuration+space+of+points'>configuration space of points</a> in <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_282' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Σ</mi></mrow><annotation encoding='application/x-tex'>\Sigma</annotation></semantics></math> may be regarded as generalized (pure) braid group. These surface braid groups are of interest in <a class='existingWikiWord' href='/nlab/show/diff/D%3D3+TQFT'>3d topological field theory</a> and in particular in <a class='existingWikiWord' href='/nlab/show/diff/topological+quantum+computation'>topological quantum computation</a> where it models <a class='existingWikiWord' href='/nlab/show/diff/braid+group+statistics'>non-abelian anyons</a>. Yet more generally, one may consider the fundamental group of the configuration space of points of any topological space <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_283' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>. For example for <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_284' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> a 1-dimensional <a class='existingWikiWord' href='/nlab/show/diff/CW+complex'>CW-complex</a>, hence an (<a class='existingWikiWord' href='/nlab/show/diff/graph'>undirected</a>) <a class='existingWikiWord' href='/nlab/show/diff/graph'>graph</a>, one speaks of graph braid groups (e.g. <a href='#FarleySabalka2009'>Farley & Sabalka 2009</a>). <blockquote> The following should maybe not be here in the Definition-section, but in some Properties- or Examples-section, or maybe in a dedicated entry on graph braid groups<a href='/nlab/new/graph+braid+groups'>?</a>: </blockquote> It has been shown (<a href='#AnMaciazek2006'>An & Maciazek 2006</a>, using discrete <a class='existingWikiWord' href='/nlab/show/diff/Morse+theory'>Morse theory</a> and combinatorial analysis of small graphs) that graph braid groups are generated by particular particle moves with the following description: <ol> <li> Star-type generators: exchanges of particle pairs on vertices of the particular graph </li> <li> loop type generators: circular moves of a single particle around a simple cycle of the graph </li> </ol> \linebreak <h4 id='AsMappingClassGroupOfAPuncturedDisk'>As mapping class group of punctured disk</h4> The braid group <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_285' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Br</mi><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Br(n)</annotation></semantics></math> may be alternatively described as the <a class='existingWikiWord' href='/nlab/show/diff/mapping+class+group'>mapping class group</a> of a 2-disk <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_286' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>D</mi> <mn>2</mn></msup></mrow><annotation encoding='application/x-tex'>D^2</annotation></semantics></math> with <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_287' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> punctures. (review includes <a href='#Birman75'>Birman 1975 §4</a>, <a href='#González-Meneses11'>González-Meneses 2011 §1.4</a>, <a href='#Abadie22'>Abadie 2022 §1.3</a>) Concretely, consider <ol> <li> <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_288' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>D</mi> <mn>2</mn></msup><mo>∖</mo><mo stretchy='false'>{</mo><msub><mi>z</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>z</mi> <mi>n</mi></msub><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>D^2 \setminus \{z_1, \cdots, z_n\}</annotation></semantics></math> denoting the <a class='existingWikiWord' href='/nlab/show/diff/complement'>complement</a> of <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_289' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> distinct points in the <a class='existingWikiWord' href='/nlab/show/diff/ball'>closed disk</a> (with <a class='existingWikiWord' href='/nlab/show/diff/boundary'>boundary</a> the <a class='existingWikiWord' href='/nlab/show/diff/circle'>circle</a>); </li> <li> <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_290' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Homeo</mi> <mo>∂</mo></msup><mo maxsize='1.2em' minsize='1.2em'>(</mo><msup><mi>D</mi> <mn>2</mn></msup><mo>∖</mo><mo stretchy='false'>{</mo><msub><mi>z</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>z</mi> <mi>n</mi></msub><mo stretchy='false'>}</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo></mrow><annotation encoding='application/x-tex'>Homeo^{\partial}\big(D^2 \setminus \{z_1, \cdots, z_n\} \big)</annotation></semantics></math> denoting the <a class='existingWikiWord' href='/nlab/show/diff/compact-open+topology'>mapping space</a> of <a class='existingWikiWord' href='/nlab/show/diff/automorphism'>auto</a>-<a class='existingWikiWord' href='/nlab/show/diff/homeomorphism'>homeomorphisms</a> which restrict to the <a class='existingWikiWord' href='/nlab/show/diff/identity'>identity</a> on the <a class='existingWikiWord' href='/nlab/show/diff/boundary'>boundary</a> <a class='existingWikiWord' href='/nlab/show/diff/circle'>circle</a>, regarded with its canonical <a class='existingWikiWord' href='/nlab/show/diff/group'>group</a> <a class='existingWikiWord' href='/nlab/show/diff/structure'>structure</a> under <a class='existingWikiWord' href='/nlab/show/diff/composition'>composition</a>; </li> <li> <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_291' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>Homeo</mi> <mi>id</mi> <mo>∂</mo></msubsup><mo maxsize='1.2em' minsize='1.2em'>(</mo><msup><mi>D</mi> <mn>2</mn></msup><mo>∖</mo><mo stretchy='false'>{</mo><msub><mi>z</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>z</mi> <mi>n</mi></msub><mo stretchy='false'>}</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo></mrow><annotation encoding='application/x-tex'>Homeo^{\partial}_id\big(D^2 \setminus \{z_1, \cdots, z_n\} \big)</annotation></semantics></math> denoting the <a class='existingWikiWord' href='/nlab/show/diff/subgroup'>subgroup</a> which is the <a class='existingWikiWord' href='/nlab/show/diff/connected+space'>connected component</a> of the <a class='existingWikiWord' href='/nlab/show/diff/identity'>identity</a> (which is readily seen to be a <a class='existingWikiWord' href='/nlab/show/diff/normal+subgroup'>normal subgroup</a>). </li> </ol> Then the <a class='existingWikiWord' href='/nlab/show/diff/mapping+class+group'>mapping class group</a> is the <a class='existingWikiWord' href='/nlab/show/diff/quotient+group'>quotient group</a>: <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_292' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>MCG</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><msup><mi>D</mi> <mn>2</mn></msup><mo>∖</mo><mo stretchy='false'>{</mo><msub><mi>z</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>z</mi> <mi>n</mi></msub><mo stretchy='false'>}</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo><mspace width='thickmathspace' /><mo>≔</mo><mspace width='thickmathspace' /><msup><mi>Homeo</mi> <mo>∂</mo></msup><mo maxsize='1.2em' minsize='1.2em'>(</mo><msup><mi>D</mi> <mn>2</mn></msup><mo>∖</mo><mo stretchy='false'>{</mo><msub><mi>z</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>z</mi> <mi>n</mi></msub><mo stretchy='false'>}</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo maxsize='1.8em' minsize='1.8em'>/</mo><msubsup><mi>Homeo</mi> <mi>id</mi> <mo>∂</mo></msubsup><mo maxsize='1.2em' minsize='1.2em'>(</mo><msup><mi>D</mi> <mn>2</mn></msup><mo>∖</mo><mo stretchy='false'>{</mo><msub><mi>z</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>z</mi> <mi>n</mi></msub><mo stretchy='false'>}</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> MCG \big( D^2 \setminus \{z_1, \cdots, z_n\} \big) \;\coloneqq\; Homeo^{\partial}\big(D^2 \setminus \{z_1, \cdots, z_n\} \big) \Big/ Homeo^{\partial}_{id}\big(D^2 \setminus \{z_1, \cdots, z_n\} \big) \,. </annotation></semantics></math></div> Now observe that <ol> <li> for the case that <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_293' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>n = 0</annotation></semantics></math> this group is <a class='existingWikiWord' href='/nlab/show/diff/trivial+group'>trivial</a>, by <a class='existingWikiWord' href='/nlab/show/diff/Alexander%27s+trick'>Alexander's trick</a>. </li> <li> continuous extension yields an <a class='existingWikiWord' href='/nlab/show/diff/injection'>injection</a> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_294' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Homeo</mi> <mo>∂</mo></msup><mo maxsize='1.2em' minsize='1.2em'>(</mo><msup><mi>D</mi> <mn>2</mn></msup><mo>∖</mo><mo stretchy='false'>{</mo><msub><mi>z</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>z</mi> <mi>n</mi></msub><mo stretchy='false'>}</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo><mover><mo>↪</mo><mrow><mphantom><mo lspace='verythinmathspace' rspace='0em'>−</mo></mphantom><mi>ι</mi><mphantom><mo lspace='verythinmathspace' rspace='0em'>−</mo></mphantom></mrow></mover><msup><mi>Homeo</mi> <mo>∂</mo></msup><mo maxsize='1.2em' minsize='1.2em'>(</mo><msup><mi>D</mi> <mn>2</mn></msup><mo maxsize='1.2em' minsize='1.2em'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> Homeo^{\partial}\big(D^2 \setminus \{z_1, \cdots, z_n\} \big) \xhookrightarrow{\phantom{-}\iota\phantom{-}} Homeo^{\partial}\big(D^2\big) \,. </annotation></semantics></math></div></li> </ol> Combining this implies that for every <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_295' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>ϕ</mi><mo stretchy='false'>]</mo><mspace width='thinmathspace' /><mo>∈</mo><mspace width='thinmathspace' /><mi>MCG</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><msup><mi>D</mi> <mn>2</mn></msup><mo>∖</mo><mo stretchy='false'>{</mo><msub><mi>z</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>z</mi> <mi>n</mi></msub><mo stretchy='false'>}</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo></mrow><annotation encoding='application/x-tex'>[\phi] \,\in\, MCG\big(D^2 \setminus \{z_1, \cdots, z_n\}\big)</annotation></semantics></math> there is an <a class='existingWikiWord' href='/nlab/show/diff/isotopy'>isotopy</a> to the <a class='existingWikiWord' href='/nlab/show/diff/identity'>identity</a>, <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_296' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ι</mi><mo stretchy='false'>[</mo><mi>ϕ</mi><mo stretchy='false'>]</mo><mo>→</mo><mi>id</mi></mrow><annotation encoding='application/x-tex'>\iota[\phi] \to id</annotation></semantics></math>, under which the locations of the punctures trace out a <a class='existingWikiWord' href='/nlab/show/diff/braid+group'>braid</a> (in the sense of a <a class='existingWikiWord' href='/nlab/show/diff/loop'>loop</a> in the symmetrized <a class='existingWikiWord' href='/nlab/show/diff/configuration+space+of+points'>configuration space of points</a>). This construction constitutes a <a class='existingWikiWord' href='/nlab/show/diff/homomorphism'>group homomorphism</a> from the <a class='existingWikiWord' href='/nlab/show/diff/mapping+class+group'>mapping class group</a> to the <a class='existingWikiWord' href='/nlab/show/diff/braid+group'>braid group</a> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_297' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>MCG</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><msup><mi>D</mi> <mn>2</mn></msup><mo>∖</mo><mo stretchy='false'>{</mo><msub><mi>z</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>z</mi> <mi>n</mi></msub><mo stretchy='false'>}</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo><mover><mo>→</mo><mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mo>∼</mo><mspace width='thickmathspace' /><mspace width='thickmathspace' /></mrow></mover><mi>Br</mi><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> MCG \big( D^2 \setminus \{z_1, \cdots, z_n\} \big) \xrightarrow{\;\;\sim\;\;} Br(n) \,. </annotation></semantics></math></div> and this is an <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphism</a>. \begin{remark}\label{ActionOfBraidGroupOnFundamentalGroupOfPuncturedDisk} (action of braid group on fundamental group of <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_298' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-punctured disk) \linebreak It follows in particular that the braid group <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_299' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Br</mi><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Br(n)</annotation></semantics></math> has a canonical <a class='existingWikiWord' href='/nlab/show/diff/action'>group action</a> on the <a class='existingWikiWord' href='/nlab/show/diff/fundamental+group'>fundamental group</a> of the <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_300' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-punctured disk. Since the latter is <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphic</a> to the <a class='existingWikiWord' href='/nlab/show/diff/free+group'>free group</a> on <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_301' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/generators+and+relations'>generators</a>, this leads to the purely algebraic characterization of the braid group <a href='#AsAutomorphismsOfAFreeGroup'>below</a>. See also eg. <a href='#MargalitWinarski21'>Margalit & Winarski (2021), §7</a>, <a href='#AmramLawrenceVishne12'>Amram, Lawrence & Vishne (2012), §7</a>. \end{remark} <h4 id='AsAutomorphismsOfAFreeGroup'>As automorphisms of a free group</h4> Since the <a class='existingWikiWord' href='/nlab/show/diff/fundamental+group'>fundamental group</a> of <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_302' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>D</mi> <mn>2</mn></msup><mo>∖</mo><mo stretchy='false'>{</mo><msub><mi>z</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>z</mi> <mi>n</mi></msub><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>D^2 \setminus \{z_1, \cdots, z_n\}</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/diff/generalized+the'>the</a> <a class='existingWikiWord' href='/nlab/show/diff/free+group'>free group</a> of <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_303' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> generators, the MCG-presentation of the braid group (<a href='#InTermsOfAutomorphismsOfFreeGroups'>above</a>) induces a <a class='existingWikiWord' href='/nlab/show/diff/homomorphism'>group homomorphism</a> of the braid group into the <a class='existingWikiWord' href='/nlab/show/diff/automorphism'>automorphism group</a> of a <a class='existingWikiWord' href='/nlab/show/diff/free+group'>free group</a>, which turns out to be <a class='existingWikiWord' href='/nlab/show/diff/faithful+representation'>faithful</a>. This presentation is due to <a href='#Artin25'>Artin 1925, §6</a>, review includes <a href='#González-Meneses11'>González-Meneses 2011, §1.6</a>, see also pointer in <a href='#Bardakov05'>Bardakov 2005, p. 2</a>. More in detail, since the <a class='existingWikiWord' href='/nlab/show/diff/homotopy+type'>homotopy type</a> of this punctured disk is, evidently, that of the the <a class='existingWikiWord' href='/nlab/show/diff/wedge+sum'>wedge sum</a> of <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_304' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/circle'>circles</a>, it follows that its <a class='existingWikiWord' href='/nlab/show/diff/fundamental+group'>fundamental group</a> is is the <a class='existingWikiWord' href='/nlab/show/diff/free+group'>free group</a> <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_305' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>F</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>F_n</annotation></semantics></math> on <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_306' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> generators: <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_307' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo maxsize='1.2em' minsize='1.2em'>(</mo><msup><mi>D</mi> <mn>2</mn></msup><mo>∖</mo><mo stretchy='false'>{</mo><msub><mi>z</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>z</mi> <mi>n</mi></msub><mo stretchy='false'>}</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><msub><mi>π</mi> <mn>1</mn></msub><mo maxsize='1.2em' minsize='1.2em'>(</mo><msub><mo>∨</mo> <mi>n</mi></msub><msup><mi>S</mi> <mn>1</mn></msup><mo maxsize='1.2em' minsize='1.2em'>)</mo><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><msub><mi>F</mi> <mi>n</mi></msub><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \pi_1 \big( D^2 \setminus \{z_1, \cdots, z_ n\} \big) \;\simeq\; \pi_1 \big( \vee_n S^1 \big) \;\simeq\; F_n \,. </annotation></semantics></math></div> Now the <a class='existingWikiWord' href='/nlab/show/diff/functor'>functoriality</a> of <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_308' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mspace width='thinmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thinmathspace' /><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy='false'>/</mo></mrow></msup><mo>⟶</mo><mi>Grp</mi></mrow><annotation encoding='application/x-tex'>\pi_1 \,\colon\, Top^{\ast/} \longrightarrow Grp</annotation></semantics></math> implies we have an induced homomorphism <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_309' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Aut</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><msup><mi>D</mi> <mn>2</mn></msup><mo>∖</mo><mo stretchy='false'>{</mo><msub><mi>z</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>z</mi> <mi>n</mi></msub><mo stretchy='false'>}</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo>⟶</mo><mi>Aut</mi><mo maxsize='1.8em' minsize='1.8em'>(</mo><msub><mi>π</mi> <mn>1</mn></msub><mo maxsize='1.2em' minsize='1.2em'>(</mo><msup><mi>D</mi> <mn>2</mn></msup><mo>∖</mo><mo stretchy='false'>{</mo><msub><mi>z</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>z</mi> <mi>n</mi></msub><mo stretchy='false'>}</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo maxsize='1.8em' minsize='1.8em'>)</mo><mspace width='thinmathspace' /><mo>≃</mo><mspace width='thinmathspace' /><mi>Aut</mi><mo stretchy='false'>(</mo><msub><mi>F</mi> <mi>n</mi></msub><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> Aut \big( D^2 \setminus \{z_1, \cdots, z_n\} \big) \longrightarrow Aut \Big( \pi_1\big( D^2 \setminus \{z_1, \cdots, z_n\} \big) \Big) \,\simeq\, Aut(F_n) \,. </annotation></semantics></math></div> If such an automorphism <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_310' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding='application/x-tex'>\phi</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_311' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>D</mi> <mn>2</mn></msup><mo>∖</mo><mo stretchy='false'>{</mo><msub><mi>z</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>z</mi> <mi>n</mi></msub><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>D^2 \setminus \{z_1, \cdots, z_n\} </annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/diff/isotopy'>isotopic</a> to the <a class='existingWikiWord' href='/nlab/show/diff/identity'>identity</a>, then of course <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_312' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy='false'>(</mo><mi>ϕ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\pi_1(\phi)</annotation></semantics></math> is trivial, which means that the above homomorphism factors through the <a class='existingWikiWord' href='/nlab/show/diff/quotient+group'>quotient group</a> known as the <a class='existingWikiWord' href='/nlab/show/diff/mapping+class+group'>mapping class group</a>: <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_313' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>MCG</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><msup><mi>D</mi> <mn>2</mn></msup><mo>∖</mo><mo stretchy='false'>{</mo><msub><mi>z</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>z</mi> <mi>n</mi></msub><mo stretchy='false'>}</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo><mspace width='thinmathspace' /><mo>=</mo><mspace width='thinmathspace' /><mi>Aut</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><msup><mi>D</mi> <mn>2</mn></msup><mo>∖</mo><mo stretchy='false'>{</mo><msub><mi>z</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>z</mi> <mi>n</mi></msub><mo stretchy='false'>}</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo stretchy='false'>/</mo><msub><mi>Aut</mi> <mn>0</mn></msub><mo maxsize='1.2em' minsize='1.2em'>(</mo><msup><mi>D</mi> <mn>2</mn></msup><mo>∖</mo><mo stretchy='false'>{</mo><msub><mi>z</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>z</mi> <mi>n</mi></msub><mo stretchy='false'>}</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo></mrow><annotation encoding='application/x-tex'> MCG \big( D^2 \setminus \{z_1, \cdots, z_n\} \big) \,=\, Aut \big( D^2 \setminus \{z_1, \cdots, z_n\} \big)/Aut_0 \big( D^2 \setminus \{z_1, \cdots, z_n\} \big) </annotation></semantics></math></div> Therefore, the above gives a homomorphism of the following form, which turns out to be a <a class='existingWikiWord' href='/nlab/show/diff/monomorphism'>monomorphism</a>: <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_314' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Br</mi><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>=</mo><mspace width='thinmathspace' /><mi>MCG</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><msup><mi>D</mi> <mn>2</mn></msup><mo>∖</mo><mo stretchy='false'>{</mo><msub><mi>z</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>z</mi> <mi>n</mi></msub><mo stretchy='false'>}</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo><mover><mo>↪</mo><mphantom><mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>−</mo></mrow></mphantom></mover><mi>Aut</mi><mo stretchy='false'>(</mo><msub><mi>F</mi> <mi>n</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> Br(n) \,=\, MCG \big( D^2 \setminus \{z_1, \cdots, z_n\} \big) \xhookrightarrow{\phantom{--}} Aut(F_n) </annotation></semantics></math></div> \begin{imagefromfile} “file_name”: “BraidActingOnFundamentalGroup-Artin1925.jpg”, “width”: 440, “float”: “right”, “unit”: “px”, “margin”: { “top”: -20, “bottom”: 20, “right”: 0, “left”: 10 }, “caption”: “from <a href='#Artin25'>Artin 1925</a>” \end{imagefromfile} Explicitly, the generator <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_315' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>yb</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>yb_i</annotation></semantics></math> <a class='maruku-eqref' href='#eq:ArtinGenerators'>(1)</a> in the Artin presentation(Def. \ref{ArtinPresentation}) is mapped to the automorphism <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_316' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>σ</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>\sigma_i</annotation></semantics></math> on the free group on <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_317' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> generators <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_318' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>t</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>t</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>t_1, \ldots, t_n</annotation></semantics></math> which is given as follows: <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_319' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>σ</mi> <mi>i</mi></msub><mo stretchy='false'>(</mo><msub><mi>t</mi> <mi>j</mi></msub><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>=</mo><mspace width='thinmathspace' /><mrow><mo>{</mo><mtable columnalign='left center left' displaystyle='false' rowspacing='0.5ex'><mtr><mtd><msub><mi>t</mi> <mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></mtd> <mtd><mo stretchy='false'>|</mo></mtd> <mtd><mi>j</mi><mo>=</mo><mi>i</mi></mtd></mtr> <mtr><mtd><msubsup><mi>t</mi> <mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msubsup><mo>⋅</mo><msub><mi>t</mi> <mi>i</mi></msub><mo>⋅</mo><msub><mi>t</mi> <mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></mtd> <mtd><mo stretchy='false'>|</mo></mtd> <mtd><mi>j</mi><mo>=</mo><mi>i</mi><mo>+</mo><mn>1</mn></mtd></mtr> <mtr><mtd><msub><mi>t</mi> <mi>j</mi></msub></mtd> <mtd><mo stretchy='false'>|</mo></mtd> <mtd><mtext>otherwise.</mtext></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \sigma_i(t_j) \,=\, \left\{ \begin{array}{lcl} t_{i+1} & \vert & j = i \\ t_{i+1}^{-1} \cdot t_i \cdot t_{i+1} & \vert & j = i + 1 \\ t_j &\vert& \text{otherwise.} \end{array} \right. </annotation></semantics></math></div> (<a href='#Artin25'>Artin 1925, (24)</a>) \begin{imagefromfile} “file_name”: “BraidActingOnFundamentalGroup-Gonzalez2011.jpg”, “width”: 540, “unit”: “px”, “margin”: { “top”: -20, “bottom”: 20, “right”: 0, “left”: 10 }, “caption”: “from <a href='#González-Meneses11'>González-Meneses 2011</a>” \end{imagefromfile} \linebreak <h2 id='properties'>Properties</h2> <h3 id='relation_to_moduli_space_of_monopoles'>Relation to moduli space of monopoles</h3> <div class='num_prop' id='ModuliSpaceOfkMonopolesStablyWeakHomotopyEquivbalentToClassifyingSpaceOfBraids'> <h6 id='proposition'>Proposition</h6> (<a class='existingWikiWord' href='/nlab/show/diff/moduli+space+of+monopoles'>moduli space of monopoles</a> is <a class='existingWikiWord' href='/nlab/show/diff/stable+weak+homotopy+equivalence'>stably weak homotopy equivalent</a> to <a class='existingWikiWord' href='/nlab/show/diff/classifying+space'>classifying space</a> of <a class='existingWikiWord' href='/nlab/show/diff/braid+group'>braid group</a>) For <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_320' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding='application/x-tex'>k \in \mathbb{N}</annotation></semantics></math> there is a <a class='existingWikiWord' href='/nlab/show/diff/stable+weak+homotopy+equivalence'>stable weak homotopy equivalence</a> between the <a class='existingWikiWord' href='/nlab/show/diff/moduli+space+of+monopoles'>moduli space of k monopoles</a> (eq:ModuliSpaceOfkInstantons) and the <a class='existingWikiWord' href='/nlab/show/diff/classifying+space'>classifying space</a> of the <a class='existingWikiWord' href='/nlab/show/diff/braid+group'>braid group</a> <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_321' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Br</mi><mo stretchy='false'>(</mo><mrow><mn>2</mn><mi>k</mi></mrow><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Br({2k})</annotation></semantics></math> on <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_322' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn><mi>k</mi></mrow><annotation encoding='application/x-tex'>2 k</annotation></semantics></math> strands: <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_323' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Σ</mi> <mn>∞</mn></msup><msub><mi>ℳ</mi> <mi>k</mi></msub><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><msup><mi>Σ</mi> <mn>∞</mn></msup><mi>Br</mi><mo stretchy='false'>(</mo><mrow><mn>2</mn><mi>k</mi></mrow><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> \Sigma^\infty \mathcal{M}_k \;\simeq\; \Sigma^\infty Br({2k}) </annotation></semantics></math></div></div> (<a href='#CohenCohenMannMilgram91'>Cohen-Cohen-Mann-Milgram 91</a>) <h2 id='Examples'>Examples</h2> The first few examples of the braid group <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_324' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Br</mi><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Br(n)</annotation></semantics></math> for low values of <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_325' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>: \begin{example} The group <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_326' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Br</mi><mo stretchy='false'>(</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Br(1)</annotation></semantics></math> has no generators and no relations, so is the <a class='existingWikiWord' href='/nlab/show/diff/trivial+group'>trivial group</a>: <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_327' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Br</mi><mo stretchy='false'>(</mo><mn>1</mn><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><mn>1</mn><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> Br(1) \;\simeq\; 1 \,. </annotation></semantics></math></div> \end{example} \begin{example} The group <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_328' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Br</mi><mo stretchy='false'>(</mo><mn>2</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Br(2)</annotation></semantics></math> has one generator and no relations, so is the infinite cyclic group of <a class='existingWikiWord' href='/nlab/show/diff/integer'>integers</a>: <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_329' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Br</mi><mo stretchy='false'>(</mo><mn>2</mn><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><mi>ℤ</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> Br(2) \;\simeq\; \mathbb{Z} \,. </annotation></semantics></math></div> \end{example} \begin{example}\label{TheBraidgroupBr3} The group <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_330' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Br</mi><mo stretchy='false'>(</mo><mn>3</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Br(3)</annotation></semantics></math> (we will simplify notation writing <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_331' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>u</mi><mo>=</mo><msub><mi>y</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>u = y_1</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_332' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>v</mi><mo>=</mo><msub><mi>y</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>v = y_2</annotation></semantics></math>) has presentation <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_333' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Br</mi><mo stretchy='false'>(</mo><mn>3</mn><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><mi>𝒫</mi><mo>≔</mo><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>:</mo><mi>r</mi><mo>≡</mo><mi>u</mi><mi>v</mi><mi>u</mi><msup><mi>v</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><msup><mi>u</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><msup><mi>v</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo>.</mo></mrow><annotation encoding='application/x-tex'> Br(3) \;\simeq\; \mathcal{P} \coloneqq \big( u,v : r \equiv u v u v^{-1} u^{-1} v^{-1} \big). </annotation></semantics></math></div> This is also known as the “trefoil <a class='existingWikiWord' href='/nlab/show/diff/knot+group'>knot group</a>”, i.e., the <a class='existingWikiWord' href='/nlab/show/diff/fundamental+group'>fundamental group</a> of the <a class='existingWikiWord' href='/nlab/show/diff/complement'>complement</a> of a <a class='existingWikiWord' href='/nlab/show/diff/trefoil+knot'>trefoil knot</a>. \end{example} \begin{example} The group <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_334' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Br</mi><mo stretchy='false'>(</mo><mn>4</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Br(4)</annotation></semantics></math> (simplifying notation as <a href='#TheBraidgroupBr3'>before</a>) has generators <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_335' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>u</mi><mo>,</mo><mi>v</mi><mo>,</mo><mi>w</mi></mrow><annotation encoding='application/x-tex'>u,v,w</annotation></semantics></math> and relations: <ul> <li> <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_336' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>r</mi> <mi>u</mi></msub><mo>≡</mo><mi>v</mi><mi>w</mi><mi>v</mi><msup><mi>w</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><msup><mi>v</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><msup><mi>w</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>r_u \equiv v w v w^{-1} v^{-1} w^{-1}</annotation></semantics></math>, </li> <li> <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_337' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>r</mi> <mi>v</mi></msub><mo>≡</mo><mi>u</mi><mi>w</mi><msup><mi>u</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><msup><mi>w</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>r_v \equiv u w u^{-1} w^{-1}</annotation></semantics></math>, </li> <li> <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_338' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>r</mi> <mi>w</mi></msub><mo>≡</mo><mi>u</mi><mi>v</mi><mi>u</mi><msup><mi>v</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><msup><mi>u</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><msup><mi>v</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>r_w \equiv u v u v^{-1} u^{-1} v^{-1}</annotation></semantics></math>. </li> </ul> \end{example} \begin{example}\label{HurwitzBraidGroup} The Hurwitz braid group (or sphere braid group) is the <a href='#ForMoreGeneralTopologicalSpaces'>surface braid group</a> for <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_339' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Σ</mi></mrow><annotation encoding='application/x-tex'>\Sigma</annotation></semantics></math> the <a class='existingWikiWord' href='/nlab/show/diff/sphere'>2-sphere</a> <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_340' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mn>2</mn></msup></mrow><annotation encoding='application/x-tex'>S^2</annotation></semantics></math>. Algebraically, the Hurwitz braid group <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_341' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>H</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding='application/x-tex'>H_{n+1}</annotation></semantics></math> has all of the generators and relations of the Artin braid group <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_342' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Br</mi><mo stretchy='false'>(</mo><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Br({n+1})</annotation></semantics></math>, plus one additional relation: <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_343' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>y</mi> <mn>1</mn></msub><msub><mi>y</mi> <mn>2</mn></msub><mi>…</mi><msub><mi>y</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><msubsup><mi>y</mi> <mi>n</mi> <mn>2</mn></msubsup><msub><mi>y</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mi>…</mi><msub><mi>y</mi> <mn>2</mn></msub><msub><mi>y</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'> y_1 y_2 \dots y_{n-1} y_n^2 y_{n-1}\dots y_2 y_1 </annotation></semantics></math></div> \end{example} <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/braid+representation'>braid representation</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/braid+group+statistics'>braid group statistics</a> <a class='existingWikiWord' href='/nlab/show/diff/braid+group+statistics'>anyons</a>, <a class='existingWikiWord' href='/nlab/show/diff/quantum+Hall+effect'>quantum Hall effect</a> <a class='existingWikiWord' href='/nlab/show/diff/topological+quantum+computation'>topological quantum computation</a> </li> <li> <a href='cryptography#BraidGroupCryptographyReferences'>braid group cryptography</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/braid+category'>braid category</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/infinitesimal+braid+relation'>infinitesimal braid relation</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/loop+braid+group'>loop braid group</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/parenthesized+braid+operad'>parenthesized braid operad</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/braid+cobordism'>braid cobordism</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/vine'>vine monoid</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/tangle'>tangle</a> </li> </ul> <table><thead><tr><th><a class='existingWikiWord' href='/nlab/show/diff/chord+diagram'>chord diagrams</a></th><th><a class='existingWikiWord' href='/nlab/show/diff/weight+system'>weight systems</a></th></tr></thead><tbody><tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/linear+chord+diagram'>linear chord diagrams</a>, <a class='existingWikiWord' href='/nlab/show/diff/chord+diagram'>round chord diagrams</a> <a class='existingWikiWord' href='/nlab/show/diff/Jacobi+diagram'>Jacobi diagrams</a>, <a class='existingWikiWord' href='/nlab/show/diff/Sullivan+chord+diagram'>Sullivan chord diagrams</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/Lie+algebra+weight+system'>Lie algebra weight systems</a>, <a class='existingWikiWord' href='/nlab/show/diff/stringy+weight+system'>stringy weight system</a>, <a class='existingWikiWord' href='/nlab/show/diff/Rozansky-Witten+weight+system'>Rozansky-Witten weight systems</a></td></tr> </tbody></table> \linebreak <table><thead><tr><th><a class='existingWikiWord' href='/nlab/show/diff/knot'>knots</a></th><th><a class='existingWikiWord' href='/nlab/show/diff/braid+group'>braids</a></th></tr></thead><tbody><tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/chord+diagram'>chord diagram</a>, <a class='existingWikiWord' href='/nlab/show/diff/Jacobi+diagram'>Jacobi diagram</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/horizontal+chord+diagram'>horizontal chord diagram</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/1T+relation'>1T</a>&<a href='4T+relation#ForCircularChordDiagrams'>4T relation</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/2T+relation'>2T</a>&<a href='4T+relation#ForHorizontalChordDiagrams'>4T relation</a>/ <a class='existingWikiWord' href='/nlab/show/diff/infinitesimal+braid+relation'>infinitesimal braid relations</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/weight+system'>weight system</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/weight+system'>horizontal weight system</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/Vassiliev+invariant'>Vassiliev knot invariant</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/Vassiliev+invariant'>Vassiliev braid invariant</a></td></tr> <tr><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/weight+systems+are+the+associated+graded+objects+of+Vassiliev+invariants'>weight systems are associated graded of Vassiliev invariants</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/weight+systems+are+cohomology+of+loop+space+of+configuration+space'>horizontal weight systems are cohomology of loop space of configuration space</a></td></tr> </tbody></table> <h2 id='references'>References</h2> <h3 id='general'>General</h3> The braid group regarded as the <a class='existingWikiWord' href='/nlab/show/diff/fundamental+group'>fundamental group</a> of a <a class='existingWikiWord' href='/nlab/show/diff/configuration+space+of+points'>configuration space of points</a> is considered (neither of them under these names, though) already in: <ul> <li id='Hurwitz1891'><a class='existingWikiWord' href='/nlab/show/diff/Adolf+Hurwitz'>Adolf Hurwitz</a>, §II of: Über Riemann’sche Flächen mit gegebenen Verzweigungspunkten, Mathematische Annalen 39 (1891) 1–60 [[doi:10.1007/BF01199469](https://doi.org/10.1007/BF01199469)]</li> </ul> there regarded as <a class='existingWikiWord' href='/nlab/show/diff/action'>acting</a> on <a class='existingWikiWord' href='/nlab/show/diff/Riemann+surface'>Riemann surfaces</a> forming <a class='existingWikiWord' href='/nlab/show/diff/branched+cover'>branched covers</a>, by movement of the branch points. The original articles dedicated to analysis of the braid group: <ul> <li id='Artin25'> <a class='existingWikiWord' href='/nlab/show/diff/Emil+Artin'>Emil Artin</a>, Theorie der Zöpfe, Abh. Math. Semin. Univ. Hambg. 4 (1925) 47–72 [[doi;10.1007/BF02950718](https://doi.org/10.1007/BF02950718)] <blockquote> (the braid group via generators & relations and via automorphisms of free groups) </blockquote> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Wilhelm+Magnus'>Wilhelm Magnus</a>, Über Automorphismen von Fundamentalgruppen berandeter Flächen, Mathematische Annalen 109 (1934) 617–646 [[doi:10.1007/BF01449158](https://doi.org/10.1007/BF01449158)] <blockquote> (the braid group as a <a class='existingWikiWord' href='/nlab/show/diff/mapping+class+group'>mapping class group</a>) </blockquote> </li> <li id='Artin47'> <a class='existingWikiWord' href='/nlab/show/diff/Emil+Artin'>Emil Artin</a>, Theory of Braids, Annals of Mathematics, Second Series, 48 1 (1947) 101-126 [[doi:10.2307/1969218](https://doi.org/10.2307/1969218)] </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Frederic+Bohnenblust'>Frederic Bohnenblust</a>, The Algebraical Braid Group, Annals of Mathematics Second Series 48 1 (1947) 127-136 [[doi:10.2307/1969219](https://doi.org/10.2307/1969219)] </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Wei-Liang+Chow'>Wei-Liang Chow</a>, On the Algebraical Braid Group, Annals of Mathematics Second Series, 49 3 (1948) 654-658 [[doi:10.2307/1969050](https://doi.org/10.2307/1969050)] </li> </ul> Survey of the early history: <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Michael+Friedman+%28historian%29'>Michael Friedman (historian)</a>, Mathematical formalization and diagrammatic reasoning: the case study of the braid group between 1925 and 1950, British Journal for the History of Mathematics 34 1 (2019) 43-59 [[doi:10.1080/17498430.2018.1533298](https://doi.org/10.1080/17498430.2018.1533298)]</li> </ul> The understanding of the braid group as the <a class='existingWikiWord' href='/nlab/show/diff/fundamental+group'>fundamental group</a> of a <a class='existingWikiWord' href='/nlab/show/diff/configuration+space+of+points'>configuration space of points</a> was re-discovered/re-vived (after <a href='#Hurwitz1891'>Hurwitz 1891</a>) in: <ul> <li id='FoxNeuwirth62'> <a class='existingWikiWord' href='/nlab/show/diff/Ralph+Fox'>Ralph H. Fox</a>, <a class='existingWikiWord' href='/nlab/show/diff/Lee+Neuwirth'>Lee Neuwirth</a>, The braid groups, Math. Scand. 10 (1962) 119-126 <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_344' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo></mrow><annotation encoding='application/x-tex'>[</annotation></semantics></math><a href='https://doi.org/10.7146/math.scand.a-10518'>doi:10.7146/math.scand.a-10518</a>, <a href='https://www.mscand.dk/article/view/10518/8539'>pdf</a>, <a href='http://www.ams.org/mathscinet-getitem?mr=150755'>MR150755</a><math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_345' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>]</annotation></semantics></math> </li> <li id='FadellNeuwirth62'> <a class='existingWikiWord' href='/nlab/show/diff/Edward+Fadell'>Edward Fadell</a>, <a class='existingWikiWord' href='/nlab/show/diff/Lee+Neuwirth'>Lee Neuwirth</a>, Configuration spaces, Math. Scand. 10 (1962) 111-118 <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_346' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo></mrow><annotation encoding='application/x-tex'>[</annotation></semantics></math><a href='https://doi.org/10.7146/math.scand.a-10517'>doi:10.7146/math.scand.a-10517</a>, <a href='http://www.ams.org/mathscinet-getitem?mr=141126'>MR141126</a><math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_347' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>]</annotation></semantics></math> </li> </ul> Textbook accounts: <ul> <li id='Birman75'> <a class='existingWikiWord' href='/nlab/show/diff/Joan+S.+Birman'>Joan S. Birman</a>, Braids, links, and mapping class groups, Princeton Univ Press (1975) [[ISBN:9780691081496](https://press.princeton.edu/books/paperback/9780691081496/braids-links-and-mapping-class-groups-am-82-volume-82), <a href='https://api.pageplace.de/preview/DT0400.9781400881420_A26691398/preview-9781400881420_A26691398.pdf'>preview pdf</a>] </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Saunders+Mac+Lane'>Saunders MacLane</a>, §XI.4 of: <a class='existingWikiWord' href='/nlab/show/diff/Categories+for+the+Working+Mathematician'>Categories for the Working Mathematician</a>, Graduate Texts in Mathematics 5 Springer (second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)] </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Tomotada+Ohtsuki'>Tomotada Ohtsuki</a> ch 2 of: Quantum Invariants – A Study of Knots, 3-Manifolds, and Their Sets, World Scientific (2001) [[doi:10.1142/4746](https://doi.org/10.1142/4746)] </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Christian+Kassel'>Christian Kassel</a>, <a class='existingWikiWord' href='/nlab/show/diff/Vladimir+Turaev'>Vladimir Turaev</a>, Braid Groups, GTM 247 Springer Heidelberg 2008 (<a href='https://link.springer.com/book/10.1007/978-0-387-68548-9'>doi:10.1007/978-0-387-68548-9</a>, <a href='http://irma.math.unistra.fr/~kassel/Braids-bk.html'>webpage</a>) </li> </ul> Further introduction and review: <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Joan+S.+Birman'>Joan S. Birman</a>, <a class='existingWikiWord' href='/nlab/show/diff/Anatoly+Libgober'>Anatoly Libgober</a> (eds.) Braids, Contemporary Mathematics 78 (1988) [[doi:10.1090/conm/078](http://dx.doi.org/10.1090/conm/078)] </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Chen+Ning+Yang'>Chen Ning Yang</a>, M. L. Ge (eds.), Braid Group, Knot Theory and Statistical Mechanics, Advanced Series in Mathematical Physics 9, World Scientific (1991) [[doi:10.1142/0796](https://doi.org/10.1142/0796)] </li> <li> Joshua Lieber, Introduction to Braid Groups, 2011 (<a href='https://math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Lieber.pdf'>pdf</a>) </li> <li id='González-Meneses11'> <a class='existingWikiWord' href='/nlab/show/diff/Juan+Gonz%C3%A1lez-Meneses'>Juan González-Meneses</a>, Basic results on braid groups , Annales Mathématiques Blaise Pascal,<del class='diffdel'> Tome</del><del class='diffdel'> 18</del><del class='diffdel'> 1</del><del class='diffdel'> (2011)</del><del class='diffdel'> 15-59</del><del class='diffdel'> [[ambp:AMBP_2011__18_1_15_0](https://ambp.centre-mersenne.org/item/?id=AMBP_2011__18_1_15_0)]</del><ins class='diffins'>18</ins><ins class='diffins'> 1 (2011) 15-59 [[ambp:AMBP_2011__18_1_15_0](https://ambp.centre-mersenne.org/item/?id=AMBP_2011__18_1_15_0), </ins><ins class='diffins'><a href='http://www.numdam.org/item/AMBP_2011__18_1_15_0'>numdam:AMBP_2011__18_1_15_0</a></ins><ins class='diffins'>]</ins> </li> <li> Alexander I. Suciu, He Wang, The pure braid groups and their relatives, Perspectives in Lie theory, 403-426, Springer INdAM series, vol. 19, Springer, Cham, 2017 (<a href='https://arxiv.org/abs/1602.05291'>arXiv:1602.05291</a>) </li> <li> Dale Rolfsen, New developments in the theory of Artin’s braid groups (<a href='https://personal.math.ubc.ca/~rolfsen/papers/newbraid/newbraid2.pdf'>pdf</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Jennifer+C.+H.+Wilson'>Jennifer C. H. Wilson</a>, The geometry and topology of braid groups, lecture at <a href='https://www.pims.math.ca/scientific-event/180611-ssgt'>2018 Summer School on Geometry and Topology</a>, Chicago (2018) [[pdf](http://www.math.lsa.umich.edu/~jchw/RTG-Braids.pdf), <a class='existingWikiWord' href='/nlab/files/Wilson-BraidGroups.pdf' title='pdf'>pdf</a>] </li> <li id='Williams20'> <a class='existingWikiWord' href='/nlab/show/diff/Lucas+Williams'>Lucas Williams</a>, Configuration Spaces for the Working Undergraduate, Rose-Hulman Undergraduate Mathematics Journal, 21 1 (2020) Article 8 [[arXiv:1911.11186](https://arxiv.org/abs/1911.11186), <a href='https://scholar.rose-hulman.edu/rhumj/vol21/iss1/8'>rhumj:vol21/iss1/8</a>] </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Jennifer+C.+H.+Wilson'>Jennifer C. H. Wilson</a>, Representation stability and the braid groups, talk at <a href='https://icerm.brown.edu/programs/sp-s22/'>ICERM – Braids</a> (Feb 2022) [[pdf](https://app.icerm.brown.edu/assets/362/3661/3661_3250_021720221430_Slides.pdf)] </li> <li id='Abadie22'> Marie Abadie, §1 in: A journey around mapping class groups and their presentations (2022)<del class='diffmod'> [[pdf](https://christianurech.github.io/Semester_Project.pdf)]</del><ins class='diffmod'> [[pdf](https://christianurech.github.io/Semester_Project.pdf),</ins><ins class='diffins'><a class='existingWikiWord' href='/nlab/files/Abadie-MCG.pdf' title='pdf'>pdf</a></ins><ins class='diffins'>]</ins> </li> </ul> See also: <ul> <li>Wikipedia: <a href='http://en.wikipedia.org/wiki/Braid_group'>Braid group</a></li> </ul> As an example of motion general “motion groups” (such as <a class='existingWikiWord' href='/nlab/show/diff/loop+braid+group'>loop braid groups</a>): <ul> <li id='Goldsmith81'>Deborah L. Goldsmith, The theory of motion groups, Michigan Math. J. 28(1): 3-17 (1981) (<a href='https://projecteuclid.org/journals/michigan-mathematical-journal/volume-28/issue-1/The-theory-of-motion-groups/10.1307/mmj/1029002454.full'>doi:10.1307/mmj/1029002454</a>)</li> </ul> Algebraic presentation of braid groups: <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Warren+Dicks'>Warren Dicks</a>, Edward Formanek, around Ex. 15.2 of: Algebraic Mapping-Class Groups of Orientable Surfaces with Boundaries, in: Infinite Groups: Geometric, Combinatorial and Dynamical Aspects, Progress in Mathematics 248 Birkhäuser (2005) [[doi :10.1007/3-7643-7447-0_4](https://doi.org/10.1007/3-7643-7447-0_4)] </li> <li> <a href='#BacarditDicks09'>Bacardit & Dicks 2009</a> </li> </ul> More <a class='existingWikiWord' href='/nlab/show/diff/finitely+presentable+group'>finite presentations</a> of the pure braid group: <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Dan+Margalit'>Dan Margalit</a>, <a class='existingWikiWord' href='/nlab/show/diff/Jon+McCammond'>Jon McCammond</a>, Geometric presentations for the pure braid group, Journal of Knot Theory and Its Ramifications 18 01 (2009) 1-20 [[arXiv:math/0603204](https://arxiv.org/abs/math/0603204), <a href='https://doi.org/10.1142/S0218216509006859'>doi:10.1142/S0218216509006859</a>] </li> <li id='Lee10'> <a class='existingWikiWord' href='/nlab/show/diff/Eon-Kyung+Lee'>Eon-Kyung Lee</a>, A positive presentation of the pure braid group, Journal of the Chungcheong Mathematical Society 23 3 (2010) 555-561 [[JAKO201007648745187](https://koreascience.kr/article/JAKO201007648745187.view?orgId=anpor), <a href='http://www.ccms.or.kr/data/pdfpaper/jcms23_3/23_3_555.pdf'>pdf</a>] </li> </ul> More on the relation of braid groups to <a class='existingWikiWord' href='/nlab/show/diff/mapping+class+group'>mapping class groups</a>: <ul> <li id='AmramLawrenceVishne12'> M. Amram, R. Lawrence, U. Vishne, Artin Covers of the Braid Group, Journal of Knot Theory and Its Ramifications 21 07 (2012) 1250061 [[doi:10.1142/S0218216512500617](https://doi.org/10.1142/S0218216512500617), <a href='http://www.ma.huji.ac.il/~ruthel/papers/12artin.pdf'>pdf</a>] </li> <li id='MargalitWinarski21'> <a class='existingWikiWord' href='/nlab/show/diff/Dan+Margalit'>Dan Margalit</a>, <a class='existingWikiWord' href='/nlab/show/diff/Rebecca+R.+Winarski'>Rebecca R. Winarski</a>, Braid groups and mapping class groups: The Birman–Hilden theory, Bull. London Math. Soc. 53 3 (2021) 643-659 [[arXiv:1703.03448](https://arxiv.org/abs/1703.03448), <a href='https://doi.org/10.1112/blms.12456'>doi:10.1112/blms.12456</a>] </li> </ul> More on the braid representation on <a class='existingWikiWord' href='/nlab/show/diff/automorphism'>automorphisms</a> of <a class='existingWikiWord' href='/nlab/show/diff/free+group'>free groups</a>: <ul> <li id='BacarditDicks09'> <a class='existingWikiWord' href='/nlab/show/diff/Llu%C3%ADs+Bacardit'>Lluís Bacardit</a>, <a class='existingWikiWord' href='/nlab/show/diff/Warren+Dicks'>Warren Dicks</a>, Actions of the braid group, and new algebraic proofs of results of Dehornoy and Larue, Groups Complexity Cryptology 1 (2009) 77-129 [[arXiv:0705.0587](https://arxiv.org/abs/0705.0587), <a href='https://doi.org/10.1515/GCC.2009.77'>doi;10.1515/GCC.2009.77</a>] </li> <li id='Bardakov05'> <a class='existingWikiWord' href='/nlab/show/diff/Valerij+G.+Bardakov'>Valerij G. Bardakov</a>, Extending representations of braid groups to the automorphism groups of free groups, Journal of Knot Theory and Its Ramifications 14 08 (2005) 1087-1098 [[arXiv:math/0408330](https://arxiv.org/abs/math/0408330), <a href='https://doi.org/10.1142/S0218216505004251'>doi:10.1142/S0218216505004251</a>] </li> <li> Tetsuya Ito, Actions of the <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_348' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-strand braid groups on the free group of rank n which are similar to the Artin representation, Quart. J. Math 66 (2015) 563-581 [[arXiv;1406.2411](https://arxiv.org/abs/1406.2411), <a href='https://doi.org/10.1093/qmath/hau033'>doi:10.1093/qmath/hau033</a>] </li> </ul> On the <a class='existingWikiWord' href='/nlab/show/diff/group+homology'>group homology</a> and <a class='existingWikiWord' href='/nlab/show/diff/group+cohomology'>group cohomology</a> of braid groups: <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Vladimir+Vershinin'>Vladimir Vershinin</a>, Homology of Braid Groups and their Generalizations, Banach Center Publications (1998) 42 1 421-446 (<a href='https://hopf.math.purdue.edu/Vershinin/hobr.pdf'>pdf</a>, <a href='https://eudml.org/doc/208821'>dml:208821</a>)</li> </ul> Relation of <a class='existingWikiWord' href='/nlab/show/diff/automorphism'>automorphism groups</a> of the <a class='existingWikiWord' href='/nlab/show/diff/profinite+completion+of+a+group'>profinite completion</a> of braid groups to the <a class='existingWikiWord' href='/nlab/show/diff/Grothendieck-Teichm%C3%BCller+tower'>Grothendieck-Teichmüller group</a>: <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Pierre+Lochak'>Pierre Lochak</a>, <a class='existingWikiWord' href='/nlab/show/diff/Leila+Schneps'>Leila Schneps</a>, The Grothendieck-Teichmüller group and automorphisms of braid groups, in: The Grothendieck Theory of Dessins d’Enfant, Cambridge University Press (1994, 2011) [[pdf](https://webusers.imj-prg.fr/~leila.schneps/Fls.pdf), <a href='https://doi.org/10.1017/CBO9780511569302'>doi:10.1017/CBO9780511569302</a>]</li> </ul> On orderings of the braid group: <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Patrick+Dehornoy'>Patrick Dehornoy</a>, Braid groups and left distributive operations , Transactions AMS 345 no.1 (1994) pp.115–150. </li> <li> H. Langmaack, Verbandstheoretische Einbettung von Klassen unwesentlich verschiedener Ableitungen in die Zopfgruppe , Computing 7 no.3-4 (1971) pp.293-310. </li> </ul> On geometric presentations of braid groups: <ul> <li>Byung Hee An, Tomasz Maciazek, Geometric Presentations of Braid Groups for Particles on a Graph, Communications in Mathematical Physics 384 (2021) 1109-1140 [[doi:10.1007/s00220-021-04095-x](http://dx.doi.org/10.1007/s00220-021-04095-x)]</li> </ul> On <a class='existingWikiWord' href='/nlab/show/diff/G-structure'>G-structure</a> for <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_349' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi><mo>=</mo><msub><mi>Br</mi> <mn>∞</mn></msub></mrow><annotation encoding='application/x-tex'>G = Br_\infty</annotation></semantics></math> the infinite braid group: <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Frederick+R.+Cohen'>Frederick R. Cohen</a>, Braid orientations and bundles with flat connections, Inventiones mathematicae 46 (1978) 99–110 [[doi:10.1007/BF01393249](https://doi.org/10.1007/BF01393249)] </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Jonathan+Beardsley'>Jonathan Beardsley</a>, On Braids and Cobordism Theories, Glasgow (2022) [notes: pdf, <a class='existingWikiWord' href='/nlab/files/Beardsley-BraidsAndCobordism.pdf' title='pdf'>pdf</a>] </li> </ul> <h3 id='braid_group_representations_as_topological_quantum_gates'>Braid group representations (as topological quantum gates)</h3> On <a class='existingWikiWord' href='/nlab/show/diff/linear+representation'>linear representations</a> of braid groups (see also at <a class='existingWikiWord' href='/nlab/show/diff/braid+group+statistics'>braid group statistics</a> and interpretation as <a class='existingWikiWord' href='/nlab/show/diff/quantum+logic+gate'>quantum gates</a> in <a class='existingWikiWord' href='/nlab/show/diff/topological+quantum+computation'>topological quantum computation</a>): <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Ivan+Marin'>Ivan Marin</a>, On the representation theory of braid groups, Annales mathématiques Blaise Pascal, 20 2 (2013) 193-260 (<a href='https://arxiv.org/abs/math/0502118'>arXiv:math/0502118</a>, <a href='https://eudml.org/doc/275607'>dml:275607</a>)</li> </ul> Review: <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Chen+Ning+Yang'>Chen Ning Yang</a>, M. L. Ge (eds.). Braid Group, Knot Theory and Statistical Mechanics, Advanced Series in Mathematical Physics 9, World Scientific (1991) <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_350' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo></mrow><annotation encoding='application/x-tex'>[</annotation></semantics></math><a href='https://doi.org/10.1142/0796'>doi:10.1142/0796</a><math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_351' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>]</annotation></semantics></math> <blockquote> (focus on <a class='existingWikiWord' href='/nlab/show/diff/quantum+Yang-Baxter+equation'>quantum Yang-Baxter equation</a>) </blockquote> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Camilo+Arias+Abad'>Camilo Arias Abad</a>, Introduction to representations of braid groups, Rev. colomb. mat. vol.49 no.1 (2015) (<a href='https://arxiv.org/abs/1404.0724'>arXiv:1404.0724</a>, <a href='https://doi.org/10.15446/recolma.v49n1.54160'>doi:10.15446/recolma.v49n1.54160</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Toshitake+Kohno'>Toshitake Kohno</a>, Introduction to representation theory of braid groups, Peking 2018 (<a href='https://www.math.pku.edu.cn/misc/puremath/summerschool/Peking_SummerSchool_kohno.pdf'>pdf</a>, <a class='existingWikiWord' href='/nlab/files/KohnoBraidRepresentations.pdf' title='pdf'>pdf</a>) </li> </ul> in relation to <a class='existingWikiWord' href='/nlab/show/diff/modular+tensor+category'>modular tensor categories</a>: <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Colleen+Delaney'>Colleen Delaney</a>, Lecture notes on modular tensor categories and braid group representations, 2019 (<a href='http://web.math.ucsb.edu/~cdelaney/MTC_Notes.pdf'>pdf</a>, <a class='existingWikiWord' href='/nlab/files/DelaneyModularTensorCategories.pdf' title='pdf'>pdf</a>)</li> </ul> Braid representations from the <a class='existingWikiWord' href='/nlab/show/diff/monodromy'>monodromy</a> of the <a class='existingWikiWord' href='/nlab/show/diff/Knizhnik-Zamolodchikov+equation'>Knizhnik-Zamolodchikov connection</a> on bundles of <a class='existingWikiWord' href='/nlab/show/diff/conformal+block'>conformal blocks</a> over <a class='existingWikiWord' href='/nlab/show/diff/configuration+space+of+points'>configuration spaces of points</a>: <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Ivan+Todorov'>Ivan Todorov</a>, <a class='existingWikiWord' href='/nlab/show/diff/Ludmil+Hadjiivanov'>Ludmil Hadjiivanov</a>, Monodromy Representations of the Braid Group, Phys. Atom. Nucl. 64 (2001) 2059-2068; Yad.Fiz. 64 (2001) 2149-2158 <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_352' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo></mrow><annotation encoding='application/x-tex'>[</annotation></semantics></math><a href='https://arxiv.org/abs/hep-th/0012099'>arXiv:hep-th/0012099</a>, <a href='https://doi.org/10.1134/1.1432899'>doi:10.1134/1.1432899</a><math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_353' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>]</annotation></semantics></math> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Ivan+Marin'>Ivan Marin</a>, Sur les représentations de Krammer génériques, Annales de l’Institut Fourier, 57 6 (2007) 1883-1925 <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_354' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo></mrow><annotation encoding='application/x-tex'>[</annotation></semantics></math><a href='http://www.numdam.org/item/AIF_2007__57_6_1883_0'>numdam:AIF_2007__57_6_1883_0</a><math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_355' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>]</annotation></semantics></math> </li> </ul> and understood in terms of <a class='existingWikiWord' href='/nlab/show/diff/braid+group+statistics'>anyon statistics</a>: <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Xia+Gu'>Xia Gu</a>, <a class='existingWikiWord' href='/nlab/show/diff/Babak+Haghighat'>Babak Haghighat</a>, <a class='existingWikiWord' href='/nlab/show/diff/Yihua+Liu'>Yihua Liu</a>, Ising- and Fibonacci-Anyons from KZ-equations, J. High Energ. Phys. 2022 15 (2022) [<a href='https://doi.org/10.1007/JHEP09(2022)015'>doi:10.1007/JHEP09(2022)015</a>, <a href='https://arxiv.org/abs/2112.07195'>arXiv:2112.07195</a>]</li> </ul> Braid representations seen inside the <a class='existingWikiWord' href='/nlab/show/diff/topological+K-theory'>topological K-theory</a> of the <a class='existingWikiWord' href='/nlab/show/diff/braid+group'>braid group</a>’s <a class='existingWikiWord' href='/nlab/show/diff/classifying+space'>classifying space</a>: <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Alejandro+Adem'>Alejandro Adem</a>, <a class='existingWikiWord' href='/nlab/show/diff/Daniel+C.+Cohen'>Daniel C. Cohen</a>, <a class='existingWikiWord' href='/nlab/show/diff/Frederick+R.+Cohen'>Frederick R. Cohen</a>, On representations and K-theory of the braid groups, Math. Ann. 326 (2003) 515-542 (<a href='https://arxiv.org/abs/math/0110138'>arXiv:math/0110138</a>, <a href='https://doi.org/10.1007/s00208-003-0435-8'>doi:10.1007/s00208-003-0435-8</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Frederick+R.+Cohen'>Frederick R. Cohen</a>, Section 3 of: On braid groups, homotopy groups, and modular forms, in: J.M. Bryden (ed.), Advances in Topological Quantum Field Theory, Kluwer 2004, 275–288 (<a href='https://link.springer.com/content/pdf/10.1007/978-1-4020-2772-7_11.pdf'>pdf</a>) </li> </ul> See also: <ul> <li>R. B. Zhang, Braid group representations arising from quantum supergroups with arbitrary <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_356' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>q</mi></mrow><annotation encoding='application/x-tex'>q</annotation></semantics></math> and link polynomials, Journal of Mathematical Physics 33, 3918 (1992) (<a href='https://doi.org/10.1063/1.529840'>doi:10.1063/1.529840</a>)</li> </ul> As <a class='existingWikiWord' href='/nlab/show/diff/quantum+logic+gate'>quantum gates</a> for <a class='existingWikiWord' href='/nlab/show/diff/topological+quantum+computation'>topological quantum computation</a> with <a class='existingWikiWord' href='/nlab/show/diff/braid+group+statistics'>anyons</a>: <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Louis+Kauffman'>Louis H. Kauffman</a>, <a class='existingWikiWord' href='/nlab/show/diff/Samuel+J.+Lomonaco'>Samuel J. Lomonaco</a>, Braiding Operators are Universal Quantum Gates, New Journal of Physics, Volume 6, January 2004 (<a href='https://arxiv.org/abs/quant-ph/0401090'>arXiv:quant-ph/0401090</a>, <a href='https://iopscience.iop.org/article/10.1088/1367-2630/6/1/134'>doi:10.1088/1367-2630/6/1/134</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Samuel+J.+Lomonaco'>Samuel J. Lomonaco</a>, <a class='existingWikiWord' href='/nlab/show/diff/Louis+Kauffman'>Louis Kauffman</a>, Topological Quantum Computing and the Jones Polynomial, Proc. SPIE 6244, Quantum Information and Computation IV, 62440Z (2006) (<a href='https://arxiv.org/abs/quant-ph/0605004'>arXiv:quant-ph/0605004</a>) <blockquote> (braid group representation serving as a <a class='existingWikiWord' href='/nlab/show/diff/topological+quantum+computation'>topological</a> <a class='existingWikiWord' href='/nlab/show/diff/quantum+logic+gate'>quantum gate</a> to compute the <a class='existingWikiWord' href='/nlab/show/diff/Jones+polynomial'>Jones polynomial</a>) </blockquote> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Louis+Kauffman'>Louis H. Kauffman</a>, <a class='existingWikiWord' href='/nlab/show/diff/Samuel+J.+Lomonaco'>Samuel J. Lomonaco</a>, Topological quantum computing and <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_357' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>SU</mi><mo stretchy='false'>(</mo><mn>2</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>SU(2)</annotation></semantics></math> braid group representations, Proceedings Volume 6976, Quantum Information and Computation VI; 69760M (2008) (<a href='https://doi.org/10.1117/12.778068'>doi:10.1117/12.778068</a>, <a href='https://www.researchgate.net/publication/228451452'>rg:228451452</a>) </li> <li> C.-L. Ho, A.I. Solomon, C.-H.Oh, Quantum entanglement, unitary braid representation and Temperley-Lieb algebra, EPL 92 (2010) 30002 (<a href='https://arxiv.org/abs/1011.6229'>arXiv:1011.6229</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Louis+Kauffman'>Louis H. Kauffman</a>, Majorana Fermions and Representations of the Braid Group, International Journal of Modern Physics AVol. 33, No. 23, 1830023 (2018) (<a href='https://arxiv.org/abs/1710.04650'>arXiv:1710.04650</a>, <a href='https://doi.org/10.1142/S0217751X18300235'>doi:10.1142/S0217751X18300235</a>) </li> <li> David Lovitz, Universal Braiding Quantum Gates [[arXiv:2304.00710](https://arxiv.org/abs/2304.00710)] </li> </ul> Introduction and review: <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Colleen+Delaney'>Colleen Delaney</a>, <a class='existingWikiWord' href='/nlab/show/diff/Eric+Rowell'>Eric C. Rowell</a>, <a class='existingWikiWord' href='/nlab/show/diff/Zhenghan+Wang'>Zhenghan Wang</a>, Local unitary representations of the braid group and their applications to quantum computing, Revista Colombiana de Matemáticas(2017), 50 (2):211 (<a href='https://arxiv.org/abs/1604.06429'>arXiv:1604.06429</a>, <a href='http://dx.doi.org/10.15446/recolma.v50n2.62211'>doi:10.15446/recolma.v50n2.62211</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Eric+Rowell'>Eric C. Rowell</a>, Braids, Motions and Topological Quantum Computing <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_358' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo></mrow><annotation encoding='application/x-tex'>[</annotation></semantics></math><a href='https://arxiv.org/abs/2208.11762'>arXiv:2208.11762</a><math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_359' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>]</annotation></semantics></math> </li> </ul> Realization of <a class='existingWikiWord' href='/nlab/show/diff/su%282%29-anyon'>Fibonacci anyons</a> on <a class='existingWikiWord' href='/nlab/show/diff/quasicrystal'>quasicrystal</a>-states: <ul> <li>Marcelo Amaral, <a class='existingWikiWord' href='/nlab/show/diff/David+Chester'>David Chester</a>, Fang Fang, Klee Irwin, Exploiting Anyonic Behavior of Quasicrystals for Topological Quantum Computing, Symmetry 14 9 (2022) 1780 <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_360' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo></mrow><annotation encoding='application/x-tex'>[</annotation></semantics></math><a href='https://arxiv.org/abs/2207.08928'>arXiv:2207.08928</a>, <a href='https://doi.org/10.3390/sym14091780'>doi:10.3390/sym14091780</a><math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_361' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>]</annotation></semantics></math></li> </ul> Realization on <a class='existingWikiWord' href='/nlab/show/diff/supersymmetry'>supersymmetric</a> <a class='existingWikiWord' href='/nlab/show/diff/spin+chain'>spin chains</a>: <ul> <li>Indrajit Jana, Filippo Montorsi, Pramod Padmanabhan, Diego Trancanelli, Topological Quantum Computation on Supersymmetric Spin Chains <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_362' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo></mrow><annotation encoding='application/x-tex'>[</annotation></semantics></math><a href='https://arxiv.org/abs/2209.03822'>arXiv:2209.03822</a><math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_363' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>]</annotation></semantics></math></li> </ul> <h3 id='ReferencesGraphBraidGroups'>Graph braid groups</h3> <ul> <li id='FarleySabalka2009'> Daniel Farley, Lucas Sabalka, Presentations of Graph Braid Groups (<a href='https://arxiv.org/abs/0907.2730'>arXiv:0907.2730</a>) </li> <li> Ki Hyoung Ko, Hyo Won Park, Characteristics of graph braid groups (<a href='https://arxiv.org/abs/1101.2648'>arXiv:1101.2648</a>) </li> <li id='AnMaciazek2006'> Byung Hee An, Tomasz Maciazek, Geometric presentations of braid groups for particles on a graph (<a href='https://arxiv.org/abs/2006.15256'>arXiv:2006.15256</a>) </li> </ul> <h3 id='relation_to_moduli_space_of_monopoles_2'>Relation to moduli space of monopoles</h3> On <a class='existingWikiWord' href='/nlab/show/diff/moduli+space+of+monopoles'>moduli spaces of monopoles</a> related to <a class='existingWikiWord' href='/nlab/show/diff/braid+group'>braid groups</a>: <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Frederick+R.+Cohen'>Fred Cohen</a>, <a class='existingWikiWord' href='/nlab/show/diff/Ralph+Cohen'>Ralph Cohen</a>, B. M. Mann, R. J. Milgram, The topology of rational functions and divisors of surfaces, Acta Math (1991) 166: 163 (<a href='https://doi.org/10.1007/BF02398886'>doi:10.1007/BF02398886</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Ralph+Cohen'>Ralph Cohen</a>, John D. S. Jones Monopoles, braid groups, and the Dirac operator, Comm. Math. Phys. Volume 158, Number 2 (1993), 241-266 (<a href='https://projecteuclid.org/euclid.cmp/1104254240'>euclid:cmp/1104254240</a>) </li> </ul> <h3 id='BraidGroupCryptographyReferences'>Braid group cryptography</h3> Partly motivated by the possibility of <a class='existingWikiWord' href='/nlab/show/diff/quantum+computation'>quantum computation</a> eventually breaking the security of <a class='existingWikiWord' href='/nlab/show/diff/cryptography'>cryptography</a> based on <a class='existingWikiWord' href='/nlab/show/diff/abelian+group'>abelian groups</a>, such as <a class='existingWikiWord' href='/nlab/show/diff/elliptic+curve'>elliptic curves</a>, there are proposals to use <a class='existingWikiWord' href='/nlab/show/diff/non-abelian+group'>non-abelian</a> <a class='existingWikiWord' href='/nlab/show/diff/braid+group'>braid groups</a> for purposes of <a class='existingWikiWord' href='/nlab/show/diff/cryptography'>cryptography</a> (“post-quantum cryptography”). <h4 id='via_conjugacy_search'>Via Conjugacy Search</h4> An early proposal was to use the Conjugacy Search Problem in <a class='existingWikiWord' href='/nlab/show/diff/braid+group'>braid groups</a> as a computationally hard problem for cryptography. This approach, though, was eventually found not to be viable. Original articles: <ul> <li> Iris Anshel, M. Anshel and D. Goldfeld, An algebraic method for public-keycryptography, Math. Research Letters 6 (1999), 287–291 (<a href='https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/1999/0006/0003/MRL-1999-0006-0003-a003.pdf'>pdf</a>) </li> <li> K.H. Ko, S.J. Lee, J.H. Cheon , J.W. Han, J. Kang, C. Park , New Public-Key Cryptosystem Using Braid Groups, In: M. Bellare (ed.) Advances in Cryptology — CRYPTO 2000 Lecture Notes in Computer Science, vol 1880. Springer 2000 (<a href='https://doi.org/10.1007/3-540-44598-6_10'>doi:10.1007/3-540-44598-6_10</a>) </li> </ul> Review: <ul> <li> Karl Mahlburg, An Overview of Braid Group Encryption, 2004 (<a href='http://www.math.wisc.edu/~boston/mahlburg.pdf'>pdf</a>) </li> <li> Parvez Anandam, Introduction to Braid Group Cryptography, 2006 (<a href='https://courses.cs.washington.edu/courses/csep590/06wi/finalprojects/anandam.pdf'>pdf</a>) </li> <li> David Garber, Braid Group Cryptography, Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore (<a href='https://arxiv.org/abs/0711.3941'>arXiv:0711.3941</a>, <a href='https://doi.org/10.1142/9789814291415_0006'>doi:10.1142/9789814291415_0006</a>) </li> <li> Cryptowiki, <a href='http://cryptowiki.net/index.php?title=Cryptosystems_based_on_braid_groups'>Cryptosystems based on braid groups</a> </li> </ul> <h4 id='via_emultiplication'>Via E-multiplication</h4> A followup proposal was to use the problem of reversing E-multiplication in braid groups, thought to remedy the previous problems. Original article: <ul> <li>Iris Anshel, Derek Atkins, Dorian Goldfeld and Paul E Gunnells, WalnutDSA(TM): A Quantum-Resistant Digital Signature Algorithm (<a href='https://eprint.iacr.org/2017/058'>eprint:2017/058</a>)</li> </ul> Review: <ul> <li>Magnus Ringerud, WalnutDSA: Another attempt at braidgroup cryptography, 2019 (<a href='https://ntnuopen.ntnu.no/ntnu-xmlui/bitstream/handle/11250/2622842/no.ntnu%3Ainspera%3A2452886.pdf?sequence=1&isAllowed=y'>pdf</a>)</li> </ul> But other problems were found with this approach, rendering it non-viable. Original article: <ul> <li>Matvei Kotov, Anton Menshov, Alexander Ushakov, An attack on the Walnut digital signature algorithm, Designs, Codes and Cryptography volume 87, pages 2231–2250 (2019) (<a href='https://doi.org/10.1007/s10623-019-00615-y'>doi:10.1007/s10623-019-00615-y</a>)</li> </ul> Review: <ul> <li>José Ignacio Escribano Pablos, María Isabel González Vasco, Misael Enrique Marriaga and Ángel Luis Pérez del Pozo, The Cracking of WalnutDSA: A Survey, in: Interactions between Group Theory, Symmetry and Cryptology, Symmetry 2019, 11(9), 1072 (<a href='https://doi.org/10.3390/sym11091072'>doi:10.3390/sym11091072</a>)</li> </ul> <h4 id='further_developments'>Further developments</h4> The basic idea is still felt to be promising: <ul> <li> Xiaoming Chen, Weiqing You, Meng Jiao, Kejun Zhang, Shuang Qing, Zhiqiang Wang, A New Cryptosystem Based on Positive Braids (<a href='https://arxiv.org/abs/1910.04346'>arXiv:1910.04346</a>) </li> <li> Garry P. Dacillo, Ronnel R. Atole, Braided Ribbon Group <math class='maruku-mathml' display='inline' id='mathml_0c867c2fb935c7a460e89d1b6bb53c6a64c6a4a8_364' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>C</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>C_n</annotation></semantics></math>-based Asymmetric Cryptography, Solid State Technology Vol. 63 No. 2s (2020) (<a href='http://www.solidstatetechnology.us/index.php/JSST/article/view/5573'>JSST:5573</a>) </li> </ul> But further attacks are being discussed: <ul> <li>James Hughes, Allen Tannenbaum, Length-Based Attacks for Certain Group Based Encryption Rewriting Systems (<a href='https://arxiv.org/abs/cs/0306032'>arXiv:cs/0306032</a>)</li> </ul> As are further ways around these: <ul> <li>Xiaoming Chen, Weiqing You, Meng Jiao, Kejun Zhang, Shuang Qing, Zhiqiang Wang, A New Cryptosystem Based on Positive Braids (<a href='https://arxiv.org/abs/1910.04346'>arXiv:1910.04346</a>)</li> </ul> <div class='property'> category: <a class='category_link' href='/nlab/list/knot+theory'>knot theory</a></div> </div> <div class="revisedby"> Last revised on October 28, 2024 at 14:07:38. 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