Results 1  10
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241
Complex reflection groups , Braid groups, Hecke algebras
, 1997
"... Presentations "a la Coxeter" are given for all (irreducible) finite complex reflection groups. They provide presentations for the corresponding generalized braid groups (for all but six cases), which allow us to generalize some of the known properties of finite Coxeter groups and their a ..."
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Cited by 183 (10 self)
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Presentations "a la Coxeter" are given for all (irreducible) finite complex reflection groups. They provide presentations for the corresponding generalized braid groups (for all but six cases), which allow us to generalize some of the known properties of finite Coxeter groups and their associated braid groups, such as the computation of the center of the braid group and the construction of deformations of the finite group algebra (Hecke algebras). We introduce monodromy representations of the braid groups which factorize through the Hecke algebras, extending results of Cherednik, Opdam, Kohno and others.
Algorithms for positive braids
 Quart. J. Math. Oxford Ser
"... We give an easily handled algorithm for the word problem in each of Artin’s braid groups, Bn, based on Garside’s methods, but framed more directly in terms of the set of positive braids in which each pair of strings crosses at most once. We develop a natural partial order on each braid group defined ..."
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Cited by 162 (1 self)
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We give an easily handled algorithm for the word problem in each of Artin’s braid groups, Bn, based on Garside’s methods, but framed more directly in terms of the set of positive braids in which each pair of strings crosses at most once. We develop a natural partial order on each braid group defined in terms of positive braids, and apply this to compare braids with different powers ∆r of the fundamental halftwist braid ∆. This leads to an improvement of Garside’s conjugacy algorithm, using a much smaller finite subset of each conjugacy class, which we term the super summit set, to represent the class, in place of Garside’s summit set.
New Publickey Cryptosystem Using Braid Groups
 Advances in cryptology—CRYPTO 2000 (Santa Barbara, CA), 166–183, Lecture Notes in Comput. Sci. 1880
, 2000
"... Abstract. The braid groups are infinite noncommutative groups naturally arising from geometric braids. The aim of this article is twofold. One is to show that the braid groups can serve as a good source to enrich cryptography. The feature that makes the braid groups useful to cryptography includes ..."
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Cited by 127 (4 self)
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Abstract. The braid groups are infinite noncommutative groups naturally arising from geometric braids. The aim of this article is twofold. One is to show that the braid groups can serve as a good source to enrich cryptography. The feature that makes the braid groups useful to cryptography includes the followings: (i) The word problem is solved via a fast algorithm which computes the canonical form which can be efficiently manipulated by computers. (ii) The group operations can be performed efficiently. (iii) The braid groups have many mathematically hard problems that can be utilized to design cryptographic primitives. The other is to propose and implement a new key agreement scheme and public key cryptosystem based on these primitives in the braid groups. The efficiency of our systems is demonstrated by their speed and information rate. The security of our systems is based on topological, combinatorial and grouptheoretical problems that are intractible according to our current mathematical knowledge. The foundation of our systems is quite different from widely used cryptosystems based on number theory, but there are some similarities in design. Key words: public key cryptosystem, braid group, conjugacy problem, key exchange, hard problem, noncommutative group, oneway function, public key infrastructure 1
A Polynomial Quantum Algorithm for Approximating the Jones Polynomial
, 2008
"... The Jones polynomial, discovered in 1984 [18], is an important knot invariant in topology. Among its many connections to various mathematical and physical areas, it is known (due to Witten [32]) to be intimately connected to Topological Quantum Field Theory (TQFT). The works of Freedman, Kitaev, Lar ..."
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Cited by 71 (3 self)
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The Jones polynomial, discovered in 1984 [18], is an important knot invariant in topology. Among its many connections to various mathematical and physical areas, it is known (due to Witten [32]) to be intimately connected to Topological Quantum Field Theory (TQFT). The works of Freedman, Kitaev, Larsen and Wang [13, 14] provide an efficient simulation of TQFT by a quantum computer, and vice versa. These results implicitly imply the existence of an efficient quantum algorithm that provides a certain additive approximation of the Jones polynomial at the fifth root of unity, e 2πi/5, and moreover, that this problem is BQPcomplete. Unfortunately, this important algorithm was never explicitly formulated. Moreover, the results in [13, 14] are heavily based on TQFT, which makes the algorithm essentially inaccessible to computer scientists. We provide an explicit and simple polynomial quantum algorithm to approximate the Jones polynomial of an n strands braid with m crossings at any primitive root of unity e 2πi/k, where the running time of the algorithm is polynomial in m, n and k. Our algorithm is based, rather than on TQFT, on well known mathematical results (specifically, the path model representation of the braid group and the uniqueness of the Markov trace for the Temperly Lieb algebra). By the results of [14], our algorithm solves a BQP complete problem. The algorithm we provide exhibits a structure which we hope is generalizable to other quantum algorithmic problems. Candidates of particular interest are the approximations of other downwards selfreducible #Phard problems, most notably, the important open problem of efficient approximation of the partition function of the Potts model, a model which is known to be tightly connected to the Jones polynomial [33].
A new approach to the conjugacy problem in Garside groups
, 2008
"... The cycling operation endows the super summit set Sx of any element x of a Garside group G with the structure of a directed graph Γx. We establish that the subset Ux of Sx consisting of the circuits of Γx can be used instead of Sx for deciding conjugacy to x in G, yielding a faster and more practica ..."
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Cited by 68 (6 self)
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The cycling operation endows the super summit set Sx of any element x of a Garside group G with the structure of a directed graph Γx. We establish that the subset Ux of Sx consisting of the circuits of Γx can be used instead of Sx for deciding conjugacy to x in G, yielding a faster and more practical solution to the conjugacy problem for Garside groups. Moreover, we present a probabilistic approach to the conjugacy search problem in Garside groups. The results are likely to have implications for the security of recently proposed cryptosystems based on the hardness of problems related to the conjugacy (search) problem in braid groups.
Conjugacy problem for braid groups and Garside groups
, 2002
"... We present a new algorithm to solve the conjugacy problem in Artin braid groups, which is faster than the one presented by Birman, Ko and Lee [3]. This algorithm can be applied not only to braid groups, but to all Garside groups (which include finite type Artin groups and torus knot groups among oth ..."
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Cited by 66 (8 self)
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We present a new algorithm to solve the conjugacy problem in Artin braid groups, which is faster than the one presented by Birman, Ko and Lee [3]. This algorithm can be applied not only to braid groups, but to all Garside groups (which include finite type Artin groups and torus knot groups among others).
A fast method for comparing braids
 Advances in Math. 125
, 1997
"... ABSTRACT. We describe a new method for comparing braid words which relies both on the automatic structure of the braid groups and on the existence of a linear ordering on braids. This syntactical algorithm is a direct generalization of the classical word reduction used in the description of free gro ..."
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Cited by 42 (12 self)
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ABSTRACT. We describe a new method for comparing braid words which relies both on the automatic structure of the braid groups and on the existence of a linear ordering on braids. This syntactical algorithm is a direct generalization of the classical word reduction used in the description of free groups, and is more efficient in practice than all previously known methods. We consider in this paper the classical braid isotopy problem, i.e., the question of deciding if a given twodimensional diagram made of a series of mutually crossing strands can be transformed into another one by moving strands but not allowing one to pass through another one. As is wellknown, this problem became a question of algebra after E. Artin in the 20’s has rephrased it as the word problem for a family of effectively presented groups, Artin’s braid groups Bn. Many solutions have been described, beginning with Artin’s original construction that uses the geometric idea of combing the braids to obtain a normal form for braid words and a decomposition of the groups Bn as semidirect products of free groups ([1]). The starting point for modern braid comparison method is the purely algebraic result by Gar
Double Loop Spaces, Braided Monoidal Categories and Algebraic 3Type of Space
 Math
, 1997
"... We show that the nerve of a braided monoidal category carries a natural action of a simplicial E2operad and is thus up to group completion a double loop space. Shifting up dimension twice associates to each braided monoidal category a 1reduced lax 3category whose nerve realizes an explicit double ..."
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Cited by 36 (2 self)
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We show that the nerve of a braided monoidal category carries a natural action of a simplicial E2operad and is thus up to group completion a double loop space. Shifting up dimension twice associates to each braided monoidal category a 1reduced lax 3category whose nerve realizes an explicit double delooping whenever all cells are invertible. We deduce that lax 3groupoids are algebraic models for homotopy 3types. Introduction The concept of braiding as a refinement of symmetry is the starting point of a rich interplay between geometry (knot theory) and algebra (representation theory). The underlying structure of a braided monoidal category reveals an interest of its own in that it encompasses two at first sight different geometrical objects : double loop spaces and homotopy 3types. The link to double loop spaces was pointed out by J. Stasheff [38] and made precise by Z. Fiedorowicz [15], who proves that double loop spaces may be characterized (up to group completion) as algebras o...
Geometric Subgroups of Surface Braid Groups
 ANN. INST. FOURIER (GRENOBLE
, 1998
"... Let M be a surface, let N be a subsurface of M , and let n m be two positive integers. The inclusion of N in M gives rise to a homomorphism from the braid group B n N with n strings on N to the braid group BmM with m strings on M . We first determine necessary and sufficient conditions that this ..."
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Cited by 35 (1 self)
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Let M be a surface, let N be a subsurface of M , and let n m be two positive integers. The inclusion of N in M gives rise to a homomorphism from the braid group B n N with n strings on N to the braid group BmM with m strings on M . We first determine necessary and sufficient conditions that this homomorphism is injective, and we characterize the commensurator, the normalizer and the centralizer of 1 N in 1 M . Then we calculate the commensurator, the normalizer, and the centralizer of B n N in BmM for large surface braid groups.