scholarly journals Attacks on the Faithfulness of the Burau Representation of the Braid Group $B_4$

2016 ◽  
Vol 8 (1) ◽  
pp. 5
Author(s):  
Mohammad Y. Chreif ◽  
Mohammad N. Abdulrahim
Keyword(s):  
Unit Circle ◽  
Complex Number ◽  
Braid Group ◽  
Simple Algorithm ◽  
Necessary Condition ◽  
Burau Representation ◽  
Open Question

<p align="left">The faithfulness of the Burau representation of the 4-strand braid group, $B_4$, remains an open question.<br />In this work, there are two main results. First, we specialize the indeterminate $t$ to a complex number on the unit circle, and we find a necessary condition for a word of $B_4$ to belong to the kernel of the representation. Second, by using a simple algorithm,<br />we will be able to exclude a family of words in the generators from belonging to the kernel of the reduced Burau representation.</p>

THE MULTI-VARIABLE ALEXANDER POLYNOMIAL OF LENS BRAIDS

2002 ◽  
Vol 11 (08) ◽  
pp. 1323-1330 ◽  
Author(s):  
NAFAA CHBILI
Keyword(s):  
Braid Group ◽  
Main Tool ◽  
Alexander Polynomial ◽  
Necessary Condition ◽  
Burau Representation ◽  
Group A

Let n ≥ 2 be an integer and ℬn the n-braid group. A braid β ∈ ℬn is said to be a (p,s)-lens braid if there exists α ∈ ℬn such that β = αp(σ1 σ2 … σn-1)ns. In this paper we use the multi-variable Alexander polynomial to find a necessary condition for a braid to be a (p,s)-lens braid, for p prime. Our main tool here is the multi-variable Burau representation of the n-braid group.

scholarly journals Irreducibility of the Tensor Product of Specializations of the Burau Representation of the Braid Groups

ISRN Algebra ◽  
2011 ◽  
Vol 2011 ◽  
pp. 1-9
Author(s):  
Mohammad N. Abdulrahim ◽  
Wiaam M. Zeid
Keyword(s):  
Tensor Product ◽  
Complex Number ◽  
Braid Group ◽  
Composition Factor ◽  
Braid Groups ◽  
Sufficient Condition ◽  
Burau Representation ◽  
Parameter Representation ◽  
Nonzero Complex Number

The reduced Burau representation is a one-parameter representation of , the braid group on strings. Specializing the parameter to nonzero complex number gives a representation : , which is either irreducible or has an irreducible composition factor : . In our paper, we let , and we determine a sufficient condition for the irreducibility of the tensor product of irreducible Burau representations. This is a generalization of our previous work concerning the cases and .

scholarly journals The Interplay between Linear Representations of the Braid Group

2007 ◽  
Vol 2007 ◽  
pp. 1-9
Author(s):  
Mohammad N. Abdulrahim ◽  
Nibal H. Kassem
Keyword(s):  
Free Group ◽  
Braid Group ◽  
Burau Representation ◽  
Standard Representation ◽  
Linear Representations ◽  
Standard Action ◽  
Composition Factors

We consider Wada's representation as a twisted version of the standard action of the braid group,Bn, on the free group withngenerators. Constructing a free group,Gnm, of ranknm, we compose Cohen's mapBn→Bnmand the embeddingBnm→Aut(Gnm)via Wada's map. We prove that the composition factors of the obtained representation are one copy of Burau representation andm−1copies of the standard representation after changing the parameterttotkin the definitions of the Burau and standard representations. This is a generalization of our previous result concerning the standard Artin representation of the braid group.

scholarly journals THE GENERALIZED MATRIX VALUED HYPERGEOMETRIC EQUATION

2010 ◽  
Vol 21 (02) ◽  
pp. 145-155 ◽  
Author(s):  
P. ROMÁN ◽  
S. SIMONDI
Keyword(s):  
Differential Equation ◽  
Orthogonal Polynomials ◽  
Unit Circle ◽  
Hypergeometric Functions ◽  
Spherical Functions ◽  
Hypergeometric Equation ◽  
Necessary Condition ◽  
Hypergeometric Differential Equation ◽  
The Matrix ◽  
Generalized Matrix

The matrix valued analog of the Euler's hypergeometric differential equation was introduced by Tirao in [4]. This equation arises in the study of matrix valued spherical functions and in the theory of matrix valued orthogonal polynomials. The goal of this paper is to extend naturally the number of parameters of Tirao's equation in order to get a generalized matrix valued hypergeometric equation. We take advantage of the tools and strategies developed in [4] to identify the corresponding matrix hypergeometric functions nFm. We prove that, if n = m + 1, these functions are analytic for |z| < 1 and we give a necessary condition for the convergence on the unit circle |z| = 1.

Coloured tangles and signatures

2017 ◽  
Vol 164 (3) ◽  
pp. 493-530 ◽  
Author(s):  
DAVID CIMASONI ◽  
ANTHONY CONWAY
Keyword(s):  
Braid Group ◽  
Maslov Index ◽  
Burau Representation

AbstractTaking the signature of the closure of a braid defines a map from the braid group to the integers. In 2005, Gambaudo and Ghys expressed the homomorphism defect of this map in terms of the Meyer cocycle and the Burau representation. In the present paper, we simultaneously extend this result in two directions, considering the multivariable signature of the closure of a coloured tangle. The corresponding defect is expressed in terms of the Maslov index and of the Lagrangian functor defined by Turaev and the first-named author.

Analogues of quantum Schubert cell algebras in PBW-deformations of quantum groups

2016 ◽  
Vol 15 (10) ◽  
pp. 1650179 ◽  
Author(s):  
Yongjun Xu ◽  
Dingguo Wang ◽  
Jialei Chen
Keyword(s):  
Weyl Group ◽  
Quantum Groups ◽  
Braid Group ◽  
Group Actions ◽  
Necessary Condition ◽  
Quantum Algebras ◽  
Ore Extension ◽  
Schubert Cell ◽  
Sufficient And Necessary Condition

We focus on a class of filtered quantum algebras [Formula: see text] which are both coideal subalgebras of quantum groups and Poincaré–Birkhoff–Witt (PBW)-deformations of their negative parts. In [Y. Xu and S. Yang, PBW-deformations of quantum groups, J. Algebra 408 (2014) 222–249], Xu and Yang proved that braid group actions on [Formula: see text] introduced by Kolb and Pellegrini can be used to define root vectors and construct PBW bases for [Formula: see text]. In this present paper, for each element [Formula: see text] in the Weyl group of [Formula: see text] we first introduce a subspace [Formula: see text] and a subalgebra [Formula: see text] of [Formula: see text], where [Formula: see text] can be considered as an analogue of quantum Schubert cell algebra. Then a sufficient and necessary condition on [Formula: see text] is given for [Formula: see text]. Moreover, we prove that [Formula: see text] if and only if [Formula: see text] and [Formula: see text] can be generated by the same simple reflections. Finally, we characterize the algebra [Formula: see text] which can be obtained via an iterated Ore extension. Our results show that quantum groups and their PBW-deformations really have some different properties.

scholarly journals EXTENDING REPRESENTATIONS OF BRAID GROUPS TO THE AUTOMORPHISM GROUPS OF FREE GROUPS

2005 ◽  
Vol 14 (08) ◽  
pp. 1087-1098 ◽  
Author(s):  
VALERIJ G. BARDAKOV
Keyword(s):  
Free Group ◽  
Braid Group ◽  
Automorphism Groups ◽  
Free Groups ◽  
Linear Representation ◽  
Braid Groups ◽  
Burau Representation ◽  
Pure Braid Group

We construct a linear representation of the group IA (Fn) of IA-automorphisms of a free group Fn, an extension of the Gassner representation of the pure braid group Pn. Although the problem of faithfulness of the Gassner representation is still open for n > 3, we prove that the restriction of our representation to the group of basis conjugating automorphisms Cbn contains a non-trivial kernel even if n = 2. We construct also an extension of the Burau representation to the group of conjugating automorphisms Cn. This representation is not faithful for n ≥ 2.

scholarly journals A free subgroup in the image of the 4-strand Burau representation

2015 ◽  
Vol 24 (12) ◽  
pp. 1550065
Author(s):  
Stefan Witzel ◽  
Matthew C. B. Zaremsky
Keyword(s):  
Free Group ◽  
Braid Group ◽  
Free Subgroup ◽  
The Other ◽  
Metric Geometry ◽  
Burau Representation ◽  
Euclidean Building

It is known that the Burau representation of the 4-strand braid group is faithful if and only if certain matrices f and k generate a (non-abelian) free group. Regarding f and k as isometries of a Euclidean building, we show that f3 and k3 generate a free group. We give two proofs, one utilizing the metric geometry of the building, and the other using simplicial retractions.

Stability analysis of fractional-order systems with double noncommensurate orders for matrix case

2011 ◽  
Vol 14 (3) ◽  
Author(s):  
Zhuang Jiao ◽  
YangQuan Chen
Keyword(s):  
Fractional Order ◽  
Linear Time ◽  
Simple Algorithm ◽  
Inverse Laplace Transform ◽  
Order System ◽  
Necessary Condition ◽  
Fractional Order Systems ◽  
Numerical Inverse Laplace Transform ◽  
Laplace Transform Technique ◽  
Matrix Case

AbstractBounded-input bounded-output stability issues for fractional-order linear time invariant (LTI) system with double noncommensurate orders for the matrix case have been established in this paper. Sufficient and necessary condition of stability is given, and a simple algorithm to test the stability for this kind of fractional-order systems is presented. Based on the numerical inverse Laplace transform technique, time-domain responses for fractional-order system with double noncommensurate orders are shown in numerical examples to illustrate the proposed results.

scholarly journals A Garside-theoretic analysis of the Burau representations

2017 ◽  
Vol 26 (07) ◽  
pp. 1750040
Author(s):  
Matthieu Calvez ◽  
Tetsuya Ito
Keyword(s):  
Braid Group ◽  
Theoretic Analysis ◽  
Burau Representation ◽  
Classical Structure ◽  
Braid Index

We establish relations between both the classical and the dual Garside structures of the braid group and the Burau representation. Using the classical structure, we formulate a non-vanishing criterion for the Burau representation of the 4-strand braid group. In the dual context, it is shown that the Burau representation for arbitrary braid index is injective when restricted to the set of simply-nested braids.

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