Braid group
Example applications of braid groups include knot theory, where any knot may be represented as the closure of certain braids (a result known as Alexander's theorem); in mathematical physics where Artin's canonical presentation of the braid group corresponds to the Yang–Baxter equation (see § Basic properties); and in monodromy invariants of algebraic geometry.The braiding of (2 + 1)-dimensional space-time trajectories formed by motion of physical rods, periodic orbits or "ghost rods", and almost-invariant sets has been used to estimate the topological entropy of several engineered and naturally occurring fluid systems, via the use of Nielsen–Thurston classification.[3][4][5] Another field of intense investigation involving braid groups and related topological concepts in the context of quantum physics is in the theory and (conjectured) experimental implementation of the proposed particles anyons.When X is the plane, the braid can be closed, i.e., corresponding ends can be connected in pairs, to form a link, i.e., a possibly intertwined union of possibly knotted loops in three dimensions.A theorem of J. W. Alexander demonstrates that every link can be obtained in this way as the "closure" of a braid.The Markov theorem gives necessary and sufficient conditions under which the closures of two braids are equivalent links.[9] Braid groups were introduced explicitly by Emil Artin in 1925, although (as Wilhelm Magnus pointed out in 1974[10]) they were already implicit in Adolf Hurwitz's work on monodromy from 1891.Braid groups may be described by explicit presentations, as was shown by Emil Artin in 1947.braid theory), an interpretation that was lost from view until it was rediscovered by Ralph Fox and Lee Neuwirth in 1962.can be abstractly defined via the following presentation: where in the first group of relationsThe image of the braid σi ∈ Bn is the transposition si = (i, i+1) ∈ Sn.This can be seen as the fundamental group of the space of n-tuples of distinct points of the Euclidean plane.The center of B3 is equal to C, a consequence of the facts that c is in the center, the modular group has trivial center, and the above surjective homomorphism has kernel C. The braid group Bn can be shown to be isomorphic to the mapping class group of a punctured disk with n punctures.Since braids can be concretely given as words in the generators σi, this is often the preferred method of entering knots into computer programs.The word problem for the braid relations is efficiently solvable and there exists a normal form for elements of Bn in terms of the generators σ1, ..., σn−1.(In essence, computing the normal form of a braid is the algebraic analogue of "pulling the strands" as illustrated in our second set of images above.)There is also a package called CHEVIE for GAP3 with special support for braid groups.[15] In analogy with the action of the symmetric group by permutations, in various mathematical settings there exists a natural action of the braid group on n-tuples of objects or on the n-folded tensor product that involves some "twists".Such structures play an important role in modern mathematical physics and lead to quantum knot invariants.Elements of the braid group Bn can be represented more concretely by matrices.It had been a long-standing question whether Burau representation was faithful, but the answer turned out to be negative for n ≥ 5.More generally, it was a major open problem whether braid groups were linear.In 1996, Chetan Nayak and Frank Wilczek posited that in analogy to projective representations of SO(3), the projective representations of the braid group have a physical meaning for certain quasiparticles in the fractional quantum hall effect.[16] Around 2001 Stephen Bigelow and Daan Krammer independently proved that all braid groups are linear.may be realized as a subgroup of the general linear group over the complex numbers.The simplest way is to take the direct limit of braid groups, where the attaching mapsdistinct unordered points in the plane:[18] So by definition The calculations for coefficients in[19] Similarly, a classifying space for the pure braid groupIn 1968 Vladimir Arnold showed that the integral cohomology of the pure braid groupis the quotient of the exterior algebra generated by the collection of degree-one classes
mathematicsn-braidsambient isotopygroup operationknot theoryAlexander's theoremmathematical physicsYang–Baxter equationmonodromyalgebraic geometryidentity elementinversefluid mechanicschaotic mixingtopological entropyNielsen–Thurston classificationquantum physicsanyonsquantum computingquantum informationhomotopyalgebraic topologyfundamental groupsconfiguration spacemanifoldsymmetric productCartesian productsymmetric grouphomotopy groupsBrunnian braidJ. W. Alexanderstring linksAndrey Markov Jr.Vaughan JonespolynomialMarkov theoremSeifert circlesEmil ArtinWilhelm MagnusAdolf Hurwitzpresentationsfundamental groupconfiguration spacesbraid theoryRalph FoxgeneratepresentationArtin groupsYang–Baxter equationstrivialcyclic groupknot grouptrefoil knotnon-abelian groupsubgrouptorsion-freelinear orderDehornoy orderfree grouphomomorphismabelianizationpermutationsurjectivegroup homomorphismCoxeter presentationkernelshort exact sequencesemi-direct productsuniversal central extensionmodular groupquotient groupcenterinner automorphismsisomorphismgeneratednormal subgroupcosetsStern–Brocot treemapping class grouppunctured diskperiodic, reducible or pseudo-Anosovword problemnormal formGAP computer algebra systemLawrence–Krammer representationcryptographytensor productinner automorphismbraided monoidal categorymonoidal categoryknot invariantsrepresentationBurau representationLaurent polynomialsfaithfullinearRuth LawrenceChetan NayakFrank Wilczekfractional quantum hall effectStephen Bigelowgeneral linear groupcomplex numbersdirect limittopologiescompletioninverse limitHilbert cubecohomology of a groupEilenberg–MacLaneclassifying spaceCW complexVladimir Arnoldexterior algebraArtin–Tits groupBraided vector spaceBraided Hopf algebraNon-commutative cryptographySpherical braid groupCommentarii Mathematici HelveticiBibcodeMarkov, AndreyRourke, Colin P.Topology and Its ApplicationsBirman, Joan S.Braids, links, and mapping class groups