An Artin Braid Group Representation of Knitting Machine State with Applications to Validation and Optimization of Fabrication Plans

2021 ◽  
Author(s):  
Jenny Lin ◽  
James McCann
Keyword(s):  
Braid Group ◽  
Group Representation ◽  
Artin Braid Group

scholarly journals METHOD OF CONSTRUCTING BRAID GROUP REPRESENTATION AND ENTANGLEMENT IN A 9 × 9 YANG–BAXTER SYSTEM

2009 ◽  
Vol 21 (09) ◽  
pp. 1081-1090 ◽  
Author(s):  
TAOTAO HU ◽  
GANGCHENG WANG ◽  
CHUNFANG SUN ◽  
CHENGCHENG ZHOU ◽  
QINGYONG WANG ◽  
...  
Keyword(s):  
Tensor Product ◽  
Braid Group ◽  
Group Representation ◽  
Standard Basis ◽  
Entangled States ◽  
Arbitrary Degree ◽  
Reducible Representation ◽  
S Matrix ◽  
Degree Of Entanglement

In this paper, we present reducible representation of the n2 braid group representation which is constructed on the tensor product of n-dimensional spaces. Specifically, it is shown that via a combining method, we can construct more n2 dimensional braiding S-matrices which satisfy the braid relations. By Yang–Baxterization approach, we derive a 9 × 9 unitary [Formula: see text]-matrix according to a 9 × 9 braiding S-matrix we have constructed. The entanglement properties of [Formula: see text]-matrix is investigated, and the arbitrary degree of entanglement for two-qutrit entangled states can be generated via [Formula: see text]-matrix acting on the standard basis.

REALIZATION OF FRT ALGEBRA RELATED TO A COLORED BRAID GROUP REPRESENTATION

1994 ◽  
Vol 09 (29) ◽  
pp. 2733-2743 ◽  
Author(s):  
B. BASU-MALLICK
Keyword(s):  
Braid Group ◽  
Group Representation ◽  
Toda Chain ◽  
Baxter Equation ◽  
Color Parameter ◽  
Triangular Matrices ◽  
Explicit Realization ◽  
Lower Triangular ◽  
Parameter Dependent ◽  
Lower Triangular Matrices

A colored braid group representation (CBGR) is constructed by using some modified universal ℛ-matrix associated with U q( gl (2)) quantized algebra. Explicit realization of Faddeev–Reshetikhin–Takhtajan (FRT) algebra, involving color parameter dependent upper and lower triangular matrices, is built up for this CBGR and subsequently applied to generate nonadditive type solutions of quantum Yang–Baxter equation. Rational limit of such solutions interestingly yields 'colored' extension of known Lax operators associated with lattice nonlinear Schrödinger model and Toda chain.

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.

Weight conservation and quantum group construction of the braid group representation

1990 ◽  
Vol 23 (13) ◽  
pp. L645-L648 ◽  
Author(s):  
Mo-Lin Ge ◽  
Chang-Pu Sun ◽  
Lu-Yu Wang ◽  
Kang Xue
Keyword(s):  
Quantum Group ◽  
Braid Group ◽  
Group Representation ◽  
Group Construction

scholarly journals THE FIBONACCI MODEL AND THE TEMPERLEY-LIEB ALGEBRA

2008 ◽  
Vol 22 (29) ◽  
pp. 5065-5080 ◽  
Author(s):  
LOUIS H. KAUFFMAN ◽  
SAMUEL J. LOMONACO
Keyword(s):  
Quantum Computation ◽  
Braid Group ◽  
Group Representation ◽  
85Th Birthday ◽  
Elementary Construction

We give an elementary construction of the Fibonacci model, a unitary braid group representation that is universal for quantum computation. This paper is dedicated to Professor C. N. Yang, on his 85th birthday.

scholarly journals VIRTUAL BRAIDS AND THE L-MOVE

2006 ◽  
Vol 15 (06) ◽  
pp. 773-811 ◽  
Author(s):  
LOUIS H. KAUFFMAN ◽  
SOFIA LAMBROPOULOU
Keyword(s):  
Braid Group ◽  
Virtual Knots ◽  
Artin Braid Group ◽  
Markov Theorem ◽  
Knots And Links ◽  
Algebraic Formulation

In this paper we prove a Markov theorem for virtual braids and for analogs of this structure including flat virtual braids and welded braids. The virtual braid group is the natural companion to the category of virtual knots, just as the Artin braid group is the natural companion to classical knots and links. In this paper we follow L-move methods to prove the Virtual Markov theorems. One benefit of this approach is a fully local algebraic formulation of the theorems in each category.

scholarly journals FUSION AND BRAIDING IN W-ALGEBRA EXTENDED CONFORMAL THEORIES (II): GENERALIZATION TO CHIRAL SCREENED VERTEX OPERATORS LABELLED BY ARBITRARY YOUNG TABLEAUX

1990 ◽  
Vol 05 (10) ◽  
pp. 1881-1909 ◽  
Author(s):  
ADEL BILAL
Keyword(s):  
Potts Model ◽  
Braid Group ◽  
Simple Form ◽  
Group Representation ◽  
Symmetric Tensor ◽  
Vertex Operators ◽  
Young Tableaux ◽  
Exchange Algebra ◽  
Form Closure ◽  
Tensor Representations

In a previous work, we defined the chiral screened vertex operators of W-algebra extended conformal theories by fusion of elementary ones. After reviewing how to obtain the braid group representation matrices, realizing the exchange algebra for those chiral vertex operators corresponding to the symmetric tensor representations of An, we generalize our results to chiral screened vertex operators associated with arbitrary An representations. The fused braiding matrices for antisymmetric tensor screened vertex operators are computed explicitly and shown to have a very simple form. Closure of the exchange algebra in the general case is proved using the relation with the Boltzmann weights of the An face models. Since, in the unitary case, the W-algebras are realized as cosets ĝk⊕ĝ1/ĝk+1, the present results can also be reinterpreted in terms of fusion of braiding matrices of the ĝ WZW models. As an example, the simplest W-algebra extended theory, the 3-state Potts model, is discussed in some detail.

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