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 Link Invariants Associated with Gauge Equivalent Solutions of the Yang–Baxter Equation: The One-Parameter Family of Minimal Typical Representations of Uq[gl(2|1)]

2003 ◽  
Vol 12 (06) ◽  
pp. 739-749
Author(s):  
Jon R. Links ◽  
David de Wit
Keyword(s):  
Braid Group ◽  
Parameter Family ◽  
Baxter Equation ◽  
Link Invariants ◽  
State Models ◽  
The One ◽  
Quantum Superalgebra

In this paper we investigate the construction of state models for link invariants using representations of the braid group obtained from various gauge choices for a solution of the trigonometric Yang–Baxter equation. Our results show that it is possible to obtain invariants of regular isotopy (as defined by Kauffman) which may not be ambient isotopic. We illustrate our results with explicit computations using solutions of the trigonometric Yang–Baxter equation associated with the one-parameter family of minimal typical representations of the quantum superalgebra Uq[gl(2|1)]. We have implemented MATHEMATICA code to evaluate the invariants for all prime knots up to 10 crossings.

scholarly journals GENERALIZED YANG–BAXTER EQUATIONS AND BRAIDING QUANTUM GATES

2012 ◽  
Vol 21 (09) ◽  
pp. 1250087 ◽  
Author(s):  
REBECCA S. CHEN
Keyword(s):  
Braid Group ◽  
Knot Theory ◽  
Information Science ◽  
Quantum Gates ◽  
Quantum Computers ◽  
Baxter Equation ◽  
Group Representations ◽  
Ongoing Research ◽  
Topological Quantum ◽  
Mathematics And Physics

Solutions to the Yang–Baxter equation — an important equation in mathematics and physics — and their afforded braid group representations have applications in fields such as knot theory, statistical mechanics, and, most recently, quantum information science. In particular, unitary representations of the braid group are desired because they generate braiding quantum gates. These are actively studied in the ongoing research into topological quantum computing. A generalized Yang–Baxter equation was proposed a few years ago by Eric Rowell et al. By finding solutions to the generalized Yang–Baxter equation, we obtain new unitary braid group representations. Our representations give rise to braiding quantum gates and thus have the potential to aid in the construction of useful quantum computers.

scholarly journals EXTRASPECIAL 2-GROUPS AND IMAGES OF BRAID GROUP REPRESENTATIONS

2006 ◽  
Vol 15 (04) ◽  
pp. 413-427 ◽  
Author(s):  
JENNIFER M. FRANKO ◽  
ERIC C. ROWELL ◽  
ZHENGHAN WANG
Keyword(s):  
Finite Groups ◽  
Braid Group ◽  
Braid Groups ◽  
Baxter Equation ◽  
Group Representations ◽  
Specific Solution ◽  
Link Invariants

We investigate a family of (reducible) representations of the braid groups [Formula: see text] corresponding to a specific solution to the Yang–Baxter equation. The images of [Formula: see text] under these representations are finite groups, and we identify them precisely as extensions of extra-special 2-groups. The decompositions of the representations into their irreducible constituents are determined, which allows us to relate them to the well-known Jones representations of [Formula: see text] factoring over Temperley–Lieb algebras and the corresponding link invariants.

scholarly journals ALL LINK INVARIANTS FOR TWO DIMENSIONAL SOLUTIONS OF YANG–BAXTER EQUATION AND DRESSINGS

2006 ◽  
Vol 15 (10) ◽  
pp. 1279-1301
Author(s):  
N. AIZAWA ◽  
M. HARADA ◽  
M. KAWAGUCHI ◽  
E. OTSUKI
Keyword(s):  
Jones Polynomial ◽  
Alexander Polynomial ◽  
Baxter Equation ◽  
Two Dimensional ◽  
Link Invariant ◽  
Three Solutions ◽  
Polynomial Invariants ◽  
Link Invariants ◽  
Higher Dimensional

All polynomial invariants of links for two dimensional solutions of Yang–Baxter equation is constructed by employing Turaev's method. As a consequence, it is proved that the best invariant so constructed is the Jones polynomial and there exist three solutions connecting to the Alexander polynomial. Invariants for higher dimensional solutions, obtained by the so-called dressings, are also investigated. It is observed that the dressings do not improve link invariant unless some restrictions are put on dressed solutions.

Strongly spin-preserving solutions of the Yang-Baxter equation and their link invariants

1993 ◽  
Vol 113 (2) ◽  
pp. 401-411
Author(s):  
M. A. Hennings
Keyword(s):  
Jones Polynomial ◽  
Alexander Polynomial ◽  
The Other ◽  
Baxter Equation ◽  
Link Invariants

Recently Kauffman[2, 3] has classified the (strongly) spin-preserving solutions of the Yang—Baxter equation, and in particular discussed two solutions. One of these can be used to obtain the Jones polynomial, while the other (with care) leads to the Alexander polynomial. In this paper we shall complete this analysis, and shall describe all strongly spin-preserving invertible solutions of the Yang-Baxter equation which lead to link invariants. Having done this, we investigate what sort of link invariants can be obtained from these solutions. It turns out that these invariants are precisely those which have been described in Hennings [l].

scholarly journals BRAID GROUP REPRESENTATIONS ARISING FROM THE YANG–BAXTER EQUATION

2010 ◽  
Vol 19 (04) ◽  
pp. 525-538 ◽  
Author(s):  
JENNIFER M. FRANKO
Keyword(s):  
Group Algebra ◽  
Braid Group ◽  
Roots Of Unity ◽  
Baxter Equation ◽  
Group Representations ◽  
Link Invariant

This paper aims to determine the images of the braid group under representations afforded by the Yang–Baxter equation when the solution is a non-trivial 4 × 4 matrix. Making the assumption that all the eigenvalues of the Yang–Baxter solution are roots of unity, leads to the conclusion that all the images are finite. Using results of Turaev, we have also identified cases in which one would get a link invariant. Finally, by observing the group algebra generated by the image of the braid group sometimes factor through known algebras, in certain instances we can identify the invariant as particular specializations of a known invariant.

SET-THEORETIC YANG–BAXTER SOLUTIONS VIA FOX CALCULUS

2006 ◽  
Vol 15 (08) ◽  
pp. 949-956 ◽  
Author(s):  
J. SCOTT CARTER ◽  
MASAHICO SAITO
Keyword(s):  
Free Group ◽  
Braid Group ◽  
Baxter Equation ◽  
Group Representations ◽  
Group Automorphisms ◽  
Free Group Automorphisms

We construct solutions to the set–theoretic Yang–Baxter equation using braid group representations in free group automorphisms and their Fox differentials. The method resembles the extensions of groups and quandles.

NEW SOLUTIONS OF THE YANG-BAXTER EQUATION AND THEIR YANG-BAXTERIZATION

1991 ◽  
Vol 06 (04) ◽  
pp. 559-576 ◽  
Author(s):  
M. COUTURE ◽  
Y. CHENG ◽  
M.L. GE ◽  
K. XUE
Keyword(s):  
Lie Algebras ◽  
Braid Group ◽  
Classical Limit ◽  
Usual Sense ◽  
Baxter Equation ◽  
Group Representations ◽  
Simple Lie Algebras

Through the examples associated with simple Lie algebras B2, C2 and D2, an approach of constructing new solutions of the spectral-independent Yang-Baxter equation (i.e. the braid group representations) was shown. These new solutions possess some particular features, such as that they do not have the classical limit in the usual sense of Refs. 2, 5, etc., and give rise to the new solutions of the x-dependent Yang-Baxter equation by using the Yang-Baxterization prescription.

STRANGE STATISTICS, BRAID GROUP REPRESENTATIONS AND MULTIPOINT FUNCTIONS IN THE N-COMPONENT MODEL

1989 ◽  
Vol 04 (09) ◽  
pp. 2333-2370 ◽  
Author(s):  
H. C. LEE ◽  
M. L. GE ◽  
M. COUTURE ◽  
Y. S. WU
Keyword(s):  
Braid Group ◽  
Point Of View ◽  
Component Model ◽  
Baxter Equation ◽  
Many Body ◽  
Group Representations ◽  
Conformal Field ◽  
Many Body Systems ◽  
Group B ◽  
Bose Einstein

The statistics of fields in low dimensions is studied from the point of view of the braid group Bn of n strings. Explicit representations MR for the N-component model, N=2 to 5, are derived by solving the Yang-Baxter-like braid group relations for the statistical matrix R, which describes the transformation of the bilinear product of two N-component fields under the transposition of coordinates. When R2≠1 the statistics is neither Bose-Einstein nor Fermi-Dirac; it is strange. It is shown that for each N, the N+1 parameter family of solutions obtained is the most general one under a given set of constraints including “charge” conservation. Extended Nth order (N>2) Alexander-Conway relations for link polynomials are derived. They depend nonhomogeneously only on one of the N+1 parameters. The N=3 and 4 ones agree with those previously derived by Akutsu et al. Flat connections ω defining integrable systems of the N-component model are derived from the representations. The monodromy of the solution of such a system also carries a representation Mω of Pn⊂Bn. For N=2, Mω=MR, but the equality may not hold in general. The connections also lead directly to solutions of the classical Yang-Baxter equation. A generalization of Kohno’s monodromy representation of Bn associated with the algebra sl(N,C) is given. Applications of the braid group representations to statistical models, conformal field theory and many-body systems of extended objects are briefly discussed.

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