scholarly journals A Basic Dimensional Representation of Artin Braid Group , and a General Burau Representation

2022 ◽  
Vol 10 (1) ◽  
pp. 145-152
Author(s):  
Arash Pourkia
Keyword(s):  
Braid Group ◽  
Dimensional Representation ◽  
Burau Representation ◽  
Artin Braid Group

Non-trivial links and plats with trivial Gassner matrices

1996 ◽  
Vol 119 (1) ◽  
pp. 43-53 ◽  
Author(s):  
Tim D. Cochran
Keyword(s):  
Normal Subgroup ◽  
Braid Group ◽  
Burau Representation ◽  
Artin Braid Group

LetBndenote the Artin braid group on ‘n-strings[ and PBnits normal subgroup consisting of all the pure braids [Bi, Mo]. These groups have been considerably scrutinized by both topologists and algebraists [BL]. One question whose answer has so far eluded us is whether or not the Gassner representationG: PBn→Mn×n(λ), into the group ofn-by-nmatrices over, is faithful (see Section 1) [Bi; ·3] [Ga]. Recently the less discriminating Burau representation B: PBn→Mn×n(Z[t±1] ) was shown to have a non-trivial kernel for each n ≥ 6 [M, LP] but these techniques have not yet yielded an element of kernel(G). This paper is a partial step in that direction.

scholarly journals On the injectivity of the Braid group in the Hecke algebra

2001 ◽  
Vol 64 (3) ◽  
pp. 487-493 ◽  
Author(s):  
Gus I. Lehrer ◽  
Nanhua Xi
Keyword(s):  
Braid Group ◽  
Hecke Algebra ◽  
Burau Representation ◽  
Coxeter System ◽  
Artin Braid Group ◽  
Coxeter Systems

We show that the well known homomorphism from any Artin braid group to the Hecke algebra of the same type is injective for the universal coxeter system and that the Burau representation is faithful for all finite coxeter systems of rank two.

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.

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.

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.

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.

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>

scholarly journals Burau Maps and Twisted Alexander Polynomials

2018 ◽  
Vol 61 (2) ◽  
pp. 479-497
Author(s):  
Anthony Conway
Keyword(s):  
Braid Group ◽  
Alexander Polynomial ◽  
Burau Representation ◽  
Alexander Polynomials

AbstractThe Burau representation of the braid group can be used to recover the Alexander polynomial of the closure of a braid. We define twisted Burau maps and use them to compute twisted Alexander polynomials.

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