Braid Group Representations Research Articles

Representations of small braid groups in Temperley–Lieb algebra

In this paper we consider the question of faithfulness of the Jones' representation of braid group B n into the Temperley–Lieb algebra TL n . The obvious motivation to study this problem is that any non-trivial element in the kernel of this representation (for any n) would almost certainly yield a non-trivial knot with trivial Jones polynomial (see [S. Bigelow, Does the Jones polynomial detect the unknot? J. Knot Theory Ramifications 11 (4) (2002) 493–505], we will explain it in more detail in Section 1). As one of the two main results we prove Theorem 1 in which we present a method to obtain non-trivial elements in the kernel of the representation of B 6 into TL 9 , 2 —to the authors' knowledge the first such examples in the second gradation of the Temperley–Lieb algebra. Theorem 2 which is a refinement of Theorem 1 may be used to produce smaller examples of the same kind. We also show briefly how some braids that are used in Section 4 to construct specific examples were generated with a computer program.

An algebra-level version of a link-polynomial identity of Lickorish

AbstractWe establish isomorphisms between certain specializations of BMW algebras and the symmetric squares of Temperley–Lieb algebras. These isomorphisms imply a link-polynomial identity due to W. B. R. Lickorish. As an application, we compute the closed images of the irreducible braid group representations factoring over these specialized BMW algebras.

Braid group representations from twisted quantum doubles of finite groups

We investigate the braid group representations arising from categories of representations of twisted quantum doubles of finite groups. For these categories, we show that the resulting braid group representations always factor through finite groups, in contrast to the categories associated with quantum groups at roots of unity. We also show that in the case of p-groups, the corresponding pure braid group representations factor through a finite p-group, which answers a question asked of the first author by V. Drinfeld.

The Jones polynomial: quantum algorithms and applications in quantum complexity theory

We analyze relationships between quantum computation and a family of generalizations of the Jones polynomial. Extending recent work by Aharonov et al., we give efficient quantum circuits for implementing the unitary Jones-Wenzl representations of the braid group. We use these to provide new quantum algorithms for approximately evaluating a family of specializations of the HOMFLYPT two-variable polynomial of trace closures of braids. We also give algorithms for approximating the Jones polynomial of a general class of closures of braids at roots of unity. Next we provide a self-contained proof of a result of Freedman et al.\ that any quantum computation can be replaced by an additive approximation of the Jones polynomial, evaluated at almost any primitive root of unity. Our proof encodes two-qubit unitaries into the rectangular representation of the eight-strand braid group. We then give QCMA-complete and PSPACE-complete problems which are based on braids. We conclude with direct proofs that evaluating the Jones polynomial of the plat closure at most primitive roots of unity is a \#P-hard problem, while learning its most significant bit is PP-hard, circumventing the usual route through the Tutte polynomial and graph coloring.

Zero Modes’ Fusion Ring and Braid Group Representations for the Extended Chiral su(2) WZNW Model

The zero modes' Fock space for the extended chiral $su(2)$ WZNW model gives room to a realization of the Grothendieck fusion ring of representations of the restricted $U_q sl(2)$ quantum universal enveloping algebra (QUEA) at an even ($2h$-th) root of unity, and of its extension by the Lusztig operators. It is shown that expressing the Drinfeld images of canonical characters in terms of Chebyshev polynomials of the Casimir invariant $C$ allows a streamlined derivation of the characteristic equation of $C$ from the defining relations of the restricted QUEA. The properties of the fusion ring of the Lusztig's extension of the QUEA in the zero modes' Fock space are related to the braiding properties of correlation functions of primary fields of the extended $su(2)_{h-2}$ current algebra model.

A trace formula for the forcing relation of braids

The forcing relation of braids has been introduced for a 2-dimensional analogue of the Sharkovskii order on periods for maps of the interval. In this paper, by making use of the Nielsen fixed point theory and a representation of braid groups, we deduce a trace formula for the computation of the forcing order.

Knot colouring polynomials

This article introduces a natural extension of colouring numbers of knots, called colouring polynomials, and studies their relationship to Yang-Baxter invariants and quandle 2-cocycle invariants. For a knot K in the 3-sphere let \pi_K be the fundamental group of the knot complement, and let (m_K,l_K) be a meridian-longitude pair in \pi_K. Given a finite group G and an element x in G, we consider the set of representations \rho from \pi_K to G that map the meridian m_K to x, and define the colouring polynomial P(K) as the sum over all longitude images \rho(l_K). The resulting invariant maps knots to the group ring Z[G]. It is multiplicative with respect to connected sum and equivariant with respect to symmetry operations of knots. Examples are given to show that colouring polynomials distinguish knots for which other invariants fail, in particular they can distinguish knots from their mutants, obverses, inverses, or reverses. We prove that every quandle 2-cocycle state-sum invariant of knots is a specialization of some knot colouring polynomial. This provides a complete topological interpretation of these invariants in terms of the knot group and its peripheral system. Furthermore, we show that P can be presented as a Yang-Baxter invariant, i.e. as the trace of some linear braid group representation. This entails in particular that Yang-Baxter invariants can detect non-inversible and non-reversible knots.

ON UNCERTAINTY, BRAIDING AND ENTANGLEMENT IN GEOMETRIC QUANTUM MECHANICS

Acting within the framework of geometric quantum mechanics, an interpretation of quantum uncertainty is discussed in terms of Jacobi fields, and a connection with the theory of elliptic curves is outlined, via classical integrability of Schrödinger's dynamics and the cross-ratio interpretation of quantum transition probabilities. Furthermore, a thoroughly geometrical construction of all special unitary representations of the 3-strand braid group on the quantum 1-qubit space is given, and the connection of one of them with elliptic curves admitting complex multiplication automorphisms — the physically relevant one corresponding to the anharmonic ratio — is shown. Also, contact is made with the Temperley–Lieb algebra theoretic constructions of Kauffman and Lomonaco, and it is shown that the standard trace relative to one of the above representations computes the Jones polynomial for particular values of the parameter, for knots arising as closures of 3-strand braids. Subsequently, a geometric entanglement criterion (in terms of Segre embeddings) is discussed, together with a projective geometrical portrait for quantum 2-gates. Finally, Aravind's idea of describing quantum states via knot theory is critically analyzed, and a geometrical picture — involving a blend of SU(2)-representation theory, classical projective geometry, binary trees and Brunnian and Hopf links — is set up in order to describe successive measurements made upon generalized GHZ states, close in spirit to the quantum knot picture again devised by Kauffman and Lomonaco.

SET-THEORETIC YANG–BAXTER SOLUTIONS VIA FOX CALCULUS

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.

A certain linear representation of the classical braid group and its application to surface braids

In two-dimensional knot theory, the relationship between surface-links and surface braids has been studied. A surface braid is described by a monodromy representation on a punctured disk which maps into the classical braid group. In this paper, we study a surface braid via a linear representation of the classical braid group and give a remarkable example of surface braids.

Universal quantum computation with theν=5∕2fractional quantum Hall state

We consider topological quantum computation (TQC) with a particular class of anyons that are believed to exist in the Fractional Quantum Hall Effect state at Landau level filling fraction nu=5/2. Since the braid group representation describing statistics of these anyons is not computationally universal, one cannot directly apply the standard TQC technique. We propose to use very noisy non-topological operations such as direct short-range interaction between anyons to simulate a universal set of gates. Assuming that all TQC operations are implemented perfectly, we prove that the threshold error rate for non-topological operations is above 14%. The total number of non-topological computational elements that one needs to simulate a quantum circuit with $L$ gates scales as $L(\log L)^3$.

On injective homomorphisms for pure braid groups, and associated Lie algebras

The purpose of this article is to record the center of the Lie algebra obtained from the descending central series of Artin's pure braid group, a Lie algebra analyzed in work of Kohno [T. Kohno, Linear representations of braid groups and classical Yang–Baxter equations, in: Contemp. Math., vol. 78, 1988, pp. 339–363; T. Kohno, Vassiliev invariants and the de Rham complex on the space of knots, in: Symplectic Geometry and Quantization, in: Contemp. Math., vol. 179, Amer. Math. Soc., Providence, RI, 1994, pp. 123–138; T. Kohno, Série de Poincaré–Koszul associée aux groupes de tresses pures, Invent. Math. 82 (1985) 57–75], and Falk and Randell [M. Falk, R. Randell, The lower central series of a fiber-type arrangement, Invent. Math. 82 (1985) 77–88]. The structure of this center gives a Lie algebraic criterion for testing whether a homomorphism out of the classical pure braid group is faithful which is analogous to a criterion used to test whether certain morphisms out of free groups are faithful [F.R. Cohen, J. Wu, On braid groups, free groups, and the loop space of the 2-sphere, in: Algebraic Topology: Categorical Decomposition Techniques, in: Progr. Math., vol. 215, Birkhäuser, Basel, 2003; Braid groups, free groups, and the loop space of the 2-sphere, math.AT/0409307]. However, it is as unclear whether this criterion for faithfulness can be applied to any open cases concerning representations of P n such as the Gassner representation.

On the kernel of the Gassner representation

We study the Gassner representation of the pure braid group P n by considering its restriction to a free subgroup F. The kernel of the restriction is shown to lie in the subgroup [Γ3F, Γ2F], sharpening a result of Lipschutz.

A REFLEXIVE REPRESENTATION OF BRAID GROUPS

In this paper, for every positive integer m, we define a representation ξn,m of the n-strand braid group Bn over a free ℤBn+m-module. It not only provides an approach to construct new representations of braid groups, but also gives a new perspective to the homological representations such as the Lawrence–Krammer representation.

Hurwitz Spaces and Braid Group Representations

Hurwitz Spaces and Braid Group Representations

Bose–Mesner algebras attached to invertible Jones pairs

In 1989, Vaughan Jones introduced spin models and showed that they could be used to form link invariants in two different ways—by constructing representations of the braid group, or by constructing partition functions. These spin models were subsequently generalized to the so-called four-weight spin models by Bannai and Bannai; these could be used to construct partition functions, but did not lead to braid group representations in any obvious way. Jaeger showed that spin models were intimately related to certain association schemes. Yamada gave a construction of a symmetric spin model on 4 n vertices from each four-weight spin model on n vertices. In this paper, we build on recent work with Munemasa to give a different proof to Yamada's result, and we analyze the structure of the association scheme attached to this spin model.

Braids and linear algebra

It is well known that braid groups act naturally on (powers of) objects of a braided monoidal category. Here we describe a braided monoidal category giving rise to braid group representations by symplectic matrices studied in [J. Assion, Math. Z. 163 (3) (1978) 291–302; B. Wajnryb, Israel J. Math. 76 (3) (1991) 265–288]. In contrast to the “standard” examples of braided monoidal categories, such as categories of representations of quantum groups, the monoidal structure in our example is given by sum of vector spaces rather than tensor product. The braiding is given by a simple formula which allows easy generalizations leading to new finite quotients of braid groups.

On representations and K-theory of the braid groups

Let Γ be the fundamental group of the complement of a K(Γ, 1) hyperplane arrangement (such as Artin's pure braid group) or more generally a homologically toroidal group as defined below. The triviality of bundles arising from orthogonal representations of Γ is characterized completely as follows. An orthogonal representation gives rise to a trivial bundle if and only if the representation factors through the spinor groups. Furthermore, the subgroup of elements in the complex K-theory of BΓ which arises from complex unitary representations of Γ is shown to be trivial. In the case of real K-theory, the subgroup of elements which arises from real orthogonal representations of Γ is an elementary abelian 2-group, which is characterized completely in terms of the first two Stiefel-Whitney classes of the representation. In addition, quadratic relations in the cohomology algebra of the pure braid groups which correspond precisely to the Jacobi identity for certain choices of Poisson algebras are shown to give the existence of certain homomorphisms from the pure braid group to generalized Heisenberg groups. These cohomology relations correspond to non-trivial Spin representations of the pure braid groups which give rise to trivial bundles.

Moduli spaces of germs of holomorphic foliations in the plane

In this paper we study the topological moduli space of some germs of singular holomorphic foliations in (C^2,0). We obtain a fully characterization for generic foliations whose vanishing order at the origin is two or three. We give a similar description for a certain subspace in the moduli space of generic germs of homogeneous foliations of any vanishing order and also for generic quasi-homogeneous foliations. In all the cases we identify the fundamental group of these spaces using the Gassner representation of the pure braid group and a suitable holonomy representation of the foliation.

Non-Abelian braid statistics versus projective permutation statistics

Recent papers by Finkelstein, Galiautdinov, and co-workers [J. Math. Phys. 42, 1489 (2001); 42, 3299 (2001)] discuss a suggestion by Wilczek that non-Abelian projective representations of the permutation group can be used as a new type of particle statistics, valid in any dimension. Wilczek’s suggestion was based in part on an analysis by Nayak and Wilczek (NW) of the non-Abelian representation of the braid group in a quantum Hall system. We point out that projective permutation statistics is not possible in a local quantum field theory as it violates locality, and show that the NW braid group representation is not equivalent to a projective representation of the permutation group. The structure of the finite image of the braid group in a 2n/2−1-dimensional representation is obtained.