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 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.

Multi-variable Burau Matrices and Labeled Line Configurations

1995 ◽  
Vol 04 (02) ◽  
pp. 235-262 ◽  
Author(s):  
Rudi Penne
Keyword(s):  
Braid Group ◽  
Pairwise Disjoint ◽  
Linear Representation ◽  
Alexander Polynomial ◽  
Path Model ◽  
Burau Representation ◽  
Line Configuration

We study labeled configurations of n pairwise disjoint lines in projective 3-space, up to “rigid isotopy”. To this end, we introduce the “Labeled Braid Group”, and give a linear representation for it, which can be regarded as a labeled version of the Burau representation. We give a topological path model for these multi-variable matrices, and use them to compute the Gassner matrix of a pure braid and the Alexander polynomial of the link associated with a labeled line configuration.

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.

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.

FERMIONS AND LINK INVARIANTS

1992 ◽  
Vol 07 (supp01a) ◽  
pp. 493-532 ◽  
Author(s):  
L. Kauffman ◽  
H. Saleur
Keyword(s):  
Braid Group ◽  
Degrees Of Freedom ◽  
Knot Theory ◽  
Alexander Polynomial ◽  
Baxter Equation ◽  
Group Representations ◽  
R Matrix ◽  
Link Invariants ◽  
Linear Representations ◽  
State Models

This paper deals with various aspects of knot theory when fermionic degrees of freedom are taken into account in the braid group representations and in the state models. We discuss how the Ř matrix for the Alexander polynomial arises from the Fox differential calculus, and how it is related to the quantum group Uqgl(1,1). We investigate new families of solutions of the Yang Baxter equation obtained from "linear" representations of the braid group and exterior algebra. We study state models associated with Uqsl(n,m), and in the case n=m=1 a state model for the multivariable Alexander polynomial. We consider invariants of links in solid handlebodies and show how the non trivial topology lifts the boson fermion degeneracy that is present in S3. We use "gauge like" changes of basis to obtain invariants in thickened surfaces Σ×[0,1].

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 Fuzzy braid group: A concept

2018 ◽  
Vol 1116 ◽  
pp. 022032
Author(s):  
M K M Nasution
Keyword(s):  
Braid Group ◽  
Group A

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.

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