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Topic: Braid group

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Symmetric group

	Braid theory - Wikipedia, the free encyclopedia
		The idea is that braids can be organised into groups, in which the group operation is 'do the first braid on a set of strings, and then follow it with a second on the twisted strings'.
		Braid groups may also be given a deeper mathematical interpretation: as the fundamental group of certain configuration spaces.
		This is invariant under the symmetric group, and Y is the quotient by the symmetric group of the non-excluded n-tuples.
	en.wikipedia.org /wiki/Braid_theory (486 words)

	PlanetMath: braid group (Site not responding. Last check: 2007-10-07)
		Homotopy corresponds to isotopy of the braid, homotopies of the strands such that none of them cross.
		The presentation was given in Artin's first paper [1] on the braid group.
		This is version 10 of braid group, born on 2003-08-15, modified 2005-03-10.
	planetmath.org /encyclopedia/Braid.html (232 words)

	List of Publications
		Braid group techniques in complex geometry V, the fundamental group of a complement of a branch curve of a Veronese generic projection, (with B. Moishezon), Communications in Analysis and Geometry 4, no. 11 (1996), 1-120.
		Braid groups, algebraic surfaces and fundamental groups of complements of branch curves, Amer.
		The fundamental group of a CP2-complement of a branch curve as an extension of a solvable group by a symmetric group, Math.
	www.cs.biu.ac.il /~teicher/publications.htm (1177 words)

	Braid group
		Braid groups were introduced explicitly by Emil Artin in 1925, although (as Wilhelm Magnus pointed out in 1974) they were already implicit in Adolf Hurwitz's work on monodromy (1891).
		The neutral element is the braid consisting of four parallel horizontal strands, and the inverse of a braid consists of that braid which "undoes" whatever the first braid did.
		The kernel of this group homomorphism is called the pure Braid group on n strands; intuitively, it consists of those braids which connect the i-th item of the left set to the i-th item of the right set, for all i.
	encycl.opentopia.com /term/Braid_group (1015 words)

	How To Braid Hair -- Recommendations and Resources (Site not responding. Last check: 2007-10-07)
		Braiding of fiber yarn creates a strand or rope that is thicker and stronger than the strands would have been separately.
		Braided streams are common wherever a drastic reduction in stream gradient causes the rapid deposition of the stream's sediment load.
		In mathematics and theoretical physics, braid statistics is a generalization of the statistics of bosons and fermions based on the concept of braid group.
	www.becomingapediatrician.com /health/76/how-to-braid-hair.html (1214 words)

	Juggling Braids (Site not responding. Last check: 2007-10-07)
		These different crossings are the generators for the braid group.
		With three ball juggling patterns, we are dealing with the braid group B
		We can analyze the braids that result from juggling patterns in terms of braid group generators to establish similarities (and differences) between these patterns.
	www.brown.edu /Students/OHJC/topology/jbraids.html (1185 words)

	Stephen Bigelow (Site not responding. Last check: 2007-10-07)
		We study the action of the braid group on the second homology of a certain configuration space by describing surfaces representing elements of homology.
		A definition of the Jones polynomial of the plat closure of a braid as an algebraic intersection pairing between elements of homology and cohomology of a certain configuration space.
		The mapping class group of a genus 2 surface is linear, with Ryan Budney.
	www.math.ucsb.edu /~bigelow/papers.html (289 words)

	Parametrized braid groups of Chevalley groups, by Jean-Louis Loday and Michael R. Stein (Site not responding. Last check: 2007-10-07)
		Parametrized braid groups of Chevalley groups, by Jean-Louis Loday and Michael R. Stein
		We introduce the notion of a braid group parametrized by a ring, which is defined by generators and relations and based on the geometric idea of painted braids.
		The technical heart of the proof is the Pure Braid Lemma, which asserts that certain elements of the parametrized braid group commute with the pure braid group.
	www.math.uiuc.edu /K-theory/0611 (185 words)

	Symmetric group - Wikipedia, the free encyclopedia
		In mathematics, the symmetric group on a set X, denoted by S
		is a group homomorphism ({+1,-1} is a group under multiplication, where +1 is e, the neutral element).
		See cube for the proper rotations of a cube, which form a group isomorphic with S
	www.wikipedia.org /wiki/Symmetric_group (480 words)

	The Circular Braid Group and Its Relationship to the Standard Braid Group (Site not responding. Last check: 2007-10-07)
		Braid theory originated in 1925 when Artin examined the relation between the braid group on n strings and the algebraic braid group.
		Geometrically, an element of a braid group on n strings is represented by n strings attached to two bars, one end of each string is attached to each bar.
		Algebraically, the braid groups themselves are represented by group generators.
	www.jyi.org /volumes/volume3/issue1/articles/singler.html (2641 words)

	Braid Group
		The identification between links, braids, and the braid group allows us to pass back and forth between the geometric study of braids (and hence knots and links) and the algebraic study of the braid group.
		Figure 5.16(a) shows a closed braid on two strands that is equivalent to the trefoil knot; Figure 5.16(b) shows a closed braid on three strands that is equivalent to the Hopf link.
		The braid relations are taken as the defining relations of the braid group.
	cnls.lanl.gov /~nbt/Book/node145.html (649 words)

	"Tree Groups and the 4 String Pure Braid Group" (Site not responding. Last check: 2007-10-07)
		In this paper we show that the unpermuted braid group on four strings is an HNN-extension of the graph group F(S), where the graph S is the complement of a path with 3 edges in the complete graph on 5 vertices.
		We will conclude, by analyzing the subgroup structure of graph groups in the case of trees, that for any tree T on a countable vertex set, F(T) is a subgroup of the 4-string braid group.
		We will also show that this uncountable collection of subgroups of the 4-string braid group is linear, that is, each subgroup embeds in GL(3,R), as well as embedding in Aut(F), where F is the free group of rank 2.
	users.wpi.edu /~hservat/tits4.html (195 words)

	Research (Site not responding. Last check: 2007-10-07)
		The first construction of the braid group was extended into larger classes of groups such as Artin groups and Garside groups, and the braid groups became immensely important in many fields, such as group theory, combinatorial group theory, topology, knot theory, computer science and algorithms, cryptography, algebraic geometry and physics.
		In the third, "Conjugacy in Artin Groups and Application to the Classification of Surfaces", we extend some results to all Artin groups of spherical type, and then give a polynomial deterministic solution which is the fastest known to us today, in spherical Artin groups in general, and in the braid group in particular.
		The generalized algorithms for computing braid monodromy together with the algorithms for solving the braid word problem may be used together as an automated method to compute the fundamental group of the complement of a given curve.
	www.cs.biu.ac.il /~kaplansh/research.htm (1160 words)

	Handweavers and Spinners Guild of Victoria Inc: Melbourne Kumihimo Group
		Braids are made using a stool or a loom and a number of weighted bobbins, the number of bobbins varying according to the style of braid to be made.
		(A braiding style dictates which order to move the bobbins in and where they are moved to.) The most versatile piece of equipment to use is the marudai (pronounced MAroo-DA-ee), a round-topped stool with a hole in the centre.
		The braids appearing in the colour display of items at the beginning of this entry are made up of 4, 8, 16 or 20 strands.
	home.vicnet.net.au /~handspin/mkgmain.htm (346 words)

	Introduction (Site not responding. Last check: 2007-10-07)
		This chapter deals with another, quite specialised, class of finitely presented groups for which the word problem is solvable, the category of braid groups.
		The word problem in braid groups is solvable, that is, there is a normal form for elements of a braid group and elements can be compared.
		There are several problems in braid groups which are believed to be mathematically hard, for example the conjugacy problem, which can be used for cryptographic purposes.
	www.umich.edu /~gpcc/scs/magma/text448.htm (551 words)

	Open Directory - Science: Math: Algebra: Group Theory (Site not responding. Last check: 2007-10-07)
		Computational Tools for Group Theory - Describes work to create a program that could be used to generate, identify, and analyze finite groups presented in the form of a Cayley Table as well as visualize the groups that are generated.
		Group Action Forum - Association for the study of the theory of transformation groups and related topics.
		Group Theory is a branch of algebra, but has strong connections with almost all parts of mathematics.
	www.dmoz.org /Science/Math/Algebra/Group_Theory (631 words)

	An Introduction to Braid Theory (Site not responding. Last check: 2007-10-07)
		This is the reason why the braid group is relevant in the study of statistics in 2-dimensional manifolds.
		This braid can be mapped to a permutation p by mapping the generators of the braid group s_i to the exchanges (i,i+1).
		The representations of the braid group have not yet been fully classified, and thus new knot invariants may arise in the future.
	www.inst.bnl.gov /~wei/braid.html (611 words)

	Category.org - The Online Shopping Center: Books - Group Theory (Site not responding. Last check: 2007-10-07)
		Subgroup growth studies the distribution of subgroups of finite index in a group as a function of the index.
		As well as determining the range of possible 'growth types', for finitely generated groups in general and for groups in particular classes such as linear groups, a main focus of the book is on the tight connection between the subgroup growth of a group and its algebraic structure.
		Throughout the emphasis is on Cayley maps: imbeddings of Cayley graphs for finite groups as (possibly branched) covering projections of surface imbeddings of loop graphs with one vertex.
	www.category.org /browse/books/13940/index.html (5868 words)

	ADVENTURERS! Gaiden - Taka no Ejiki (Site not responding. Last check: 2007-10-07)
		But apparently fear of what their superiors would do if the Army group came back empty-handed was greater than their fear of what the archer might be capable of, and the goons inched towards the girl, closing the circle gradually as they went.
		The braid was held in place by an odd hair band near its end, which had a large number of feathers identical to those on the arrows that were scattered around the clearing attached to its inside surface.
		A larger group of Khrima goons was now gathered there, apparently waiting for something to happen, or maybe psyching themselves up to go and take care of the monsters they thought were about to attack before the creatures came into the village.
	www.rpi.edu /~bonnes/ADVGaidenFinal.htm (15760 words)

	Finite-Dimensional Representations of Artin's Braid Group. - Birman, Long, Moody (ResearchIndex) (Site not responding. Last check: 2007-10-07)
		The earliest known non obvious representation of the braid groups was the Burau representation, de ned by Burau [6] in 1935.
		4 The automorphism group of a free group is not linear (context) - Formanek, Procesi
		3 the linearity of the automorphism groups of free groups (context) - Dyer, Formanek et al.
	citeseer.ist.psu.edu /birman94finitedimensional.html (612 words)

	Accessing Information (Site not responding. Last check: 2007-10-07)
		Given a braid group B, return the number of Artin generators of B. Note the returned value is one less than the number of strings used in the braid group constructor BraidGroup.
		Given an element of a braid group, represented as word u in the Artin generators, return a list of integers corresponding to the generator indices in u.
		Given an element u of a braid group B on n strings, return the parameters describing the normal form of u as outlined in Section Normal Form for Elements of a Braid Group.
	www.umich.edu /~gpcc/scs/magma/text450.htm (440 words)

	PlanetMath: (Site not responding. Last check: 2007-10-07)
		A groups embeds into its profinite completion if and only if it is residually finite owned by avf
		alternating group is a normal subgroup of the symmetric group owned by mathcam
		Artin's braid group (=braid group) owned by bwebste
	planetmath.org /encyclopedia/A (1757 words)

	Self-Styled "Orders of Saint John" II (Site not responding. Last check: 2007-10-07)
		Stefanizzi continued to run his group separately and, on June 9, 1986, wrote to President Aquino of the Philippines in an attempt to establish mutual diplomatic relations and naming a Mr Antonio Delgado to the rank of Ambassador; unfortunately, as the SMHOM already had diplomatic relations with the Republic, he was unsuccessful.
		Nonetheless, the Kansas City group evidently consider that Valitch confers something they need upon their organization because in one of their booklets they reproduced a photo of Valitch in clerical robes at a public audience with his Holiness, in which the Pope appears to be rejecting something Valitch is offering.
		Associated as officers of this group were Joseph A. Storace ("grand bailiff"), Godwin Drago "grand marshal", John Wilkinson "prince of Badenberg" ("grand prior of America"), and Richard D. Murray, MD, of Ohio as Registrar and Acting-Treasurer, Tonna-Barthet himself being variously "grand chancellor" and "grand commander".
	www.chivalricorders.org /orders/self-styled/selfsty2.htm (12539 words)

	[No title]
		Also, even though a braid from the Artin canonical factors and a braid from the band canonical factors may map to the same permutation, they are not necessarily the same braid.
		Finally, the input braid represented by [3 2 1] is the fundamental braid in the Artin generators.
		The braid represented by the output [3 2 3] maps to the same permutation, but the strings are interlaced in a different way.
	www.math.wisc.edu /~boston/bolstad.doc (1585 words)

	Citations: New Key Agreement Protocols in Braid Group Cryptography - Anshel, Anshel, Fisher, Goldfeld (ResearchIndex) (Site not responding. Last check: 2007-10-07)
		In both cryptosystems a braid group B n is used as the platform.
		We are not going into any technical details about braid groups here since they are irrelevant to the subject of the present....
		The method is based on having a canonical minimal length form for words in a given finitely presented group, which can be computed rather rapidly, and in which there is no corresponding fast solution for the conjugacy problem.
	citeseer.ist.psu.edu /context/2056563/0 (1015 words)

	Algorithmic and cohomological aspects of Aut(Fn) and its subgroups (Site not responding. Last check: 2007-10-07)
		Normal forms for the group of basis-conjugating automorphisms of a free group (with M. Gutierrez), submitted.
		Bridson and K. Vogtmann, On the geometry of the automorphism froup of a free group, preprint 1994.
		Brownstein and R. Lee, Cohomology of the group of motions of nstrings in the 3-space, Contemp.
	www.cse.ogi.edu /~krstic/summary/node3.html (325 words)

	[No title]
		The braid group platform we have in mind requires some modification of the commutator key exchange protocol previously discussed.
		This matrix group consists certain of matrices of dimension N=n(n+1)/2 whose entries are represented by finite Laurent series in two variables.
		Employ LK-CKE with suitable choice of braid group parameters such that the number of elementary field operations required to solve the associated systems of linear equations is at least O(10^36).
	www-cs.engr.ccny.cuny.edu /~csmma/mmabgcqc (1103 words)

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