Burau Representation Research Articles

Generators for the hyperelliptic Torelli group and the kernel of the Burau representation at $$t=-1$$ t = - 1

We prove that the hyperelliptic Torelli group is generated by Dehn twists about separating curves that are preserved by the hyperelliptic involution. This verifies a conjecture of Hain. The hyperelliptic Torelli group can be identified with the kernel of the Burau representation evaluated at t=-1 and also the fundamental group of the branch locus of the period mapping, and so we obtain analogous generating sets for those. One application is that each component in Torelli space of the locus of hyperelliptic curves becomes simply connected when curves of compact type are added.

Braiding fluxes in Pauli Hamiltonian

Aharonov and Casher showed that Pauli Hamiltonians in two dimensions have gapless zero modes. We study the adiabatic evolution of these modes under the slow motion of N fluxons with fluxes Φa∈R. The positions, ra∈R2, of the fluxons are viewed as controls. We are interested in the holonomies associated with closed paths in the space of controls. The holonomies can sometimes be abelian, but in general are not. They can sometimes be topological, but in general are not. We analyse some of the special cases and some of the general ones. Our most interesting results concern the cases where holonomy turns out to be topological which is the case when all the fluxons are subcritical, Φa<1, and the number of zero modes is D=N−1. If N≥3 it is also non-abelian. In the special case that the fluxons carry identical fluxes the resulting anyons satisfy the Burau representations of the braid group.

Image of the Burau representation at d-th roots of unity

We prove that the image of the Full braid group $B_{n+1}$ on $n+1$ strands under the Burau representation, evaluated at a primitive $d$-th root of unity is arithmetic provided $n\geq d$.

Point pushing, homology, and the hyperelliptic involution

The hyperelliptic Torelli group is the subgroup of the mapping class group consisting of elements that act trivially on the homology of the surface and that also commute with some fixed hyperelliptic involution. We prove a Birman exact sequence for hyperelliptic Torelli groups, and we show that this sequence splits. As a consequence, we show that the hyperelliptic Torelli group is generated by Dehn twists if and only if it is generated by reducible elements. We also give an application to the kernel of the Burau representation.

CATEGORICAL ASPECTS OF VIRTUALITY AND SELF-DISTRIBUTIVITY

This paper revolves around two main results. First, we propose a "hom-set" type categorification of virtual braid groups and positive virtual braid monoids in terms of "locally" braided objects in a symmetric category (SC). This "double braiding" approach provides a rich source of representations, and offers a natural categorical interpretation for virtual racks and for the twisted Burau representation. Second, we define self-distributive (SD) structures in an arbitrary SC. SD structures are shown to produce braided objects in a SC. As for examples, we interpret the associativity and the Jacobi identity in a SC as generalized self-distributivity, thus endowing associative and Leibniz algebras with a (pre-)braiding. A homology theory of categorical SD structures is developed using the "braided" techniques from [Lebed, Homologies of algebraic structures via braidings and quantum shuffles, to appear in J. Algebra], generalizing rack, bar, Leibniz and other familiar complexes.

Local representations of braid groups

We study the local linear representations of the braid group B 3 and the homogeneous local representations of B n for n ≥ 2. We investigate the connection of these representations with the Burau representation. The linear representations of B n are constructed from the Wada representation of B n in the automorphism group of a free group.

Security analysis of key agreement protocol based on matrix power function

Key agreement protocol (KAP) using Burau braid groups representation and matrix power function (MPF) is analyzed. MPF arguments are Burau representation matrices defined over finite field or ring. It is shown that KAP security relies on the solution of matrix multivariate quadratic system of equations over the ring with additional commutation constraints for matrices to be found. We are making a conjecture that proposed KAP is a candidate one-way function since its inversion is related with the solution of known multivariate quadratic problem which is NP-complete over any field. The one of advantages of proposed KAP is its possible effective realization even in restricted computational environments by avoiding arithmetic operations with big integers.

Cohomology of the hyperelliptic Torelli group

Let SI(Sg) denote the hyperelliptic Torelli group of a closed surface Sg of genus g. This is the subgroup of the mapping class group of Sg consisting of elements that act trivially on H1(Sg; ℤ) and that commute with some fixed hyperelliptic involution of Sg. We prove that the cohomological dimension of SI(Sg) is g − 1 when g ≥ 1. We also show that Hg−1(SI(Sg); ℤ) is infinitely generated when g ≥ 2. In particular, SI(S3) is not finitely presentable. Finally, we apply our main results to show that the kernel of the Burau representation of the braid group Bn at t = −1 has cohomological dimension equal to the integer part of n/2, and it has infinitely generated homology in this top dimension.

Irreducibility of the Tensor Product of Specializations of the Burau Representation of the Braid Groups

The reduced Burau representation is a one-parameter representation of , the braid group on strings. Specializing the parameter to nonzero complex number gives a representation : , which is either irreducible or has an irreducible composition factor : . In our paper, we let , and we determine a sufficient condition for the irreducibility of the tensor product of irreducible Burau representations. This is a generalization of our previous work concerning the cases and .

A family of representations of braid groups on surfaces

We propose a family of new representations of the braid groups on surfaces that extend linear representations of the braid groups on a disc such as the Burau representation and the Lawrence-Krammer-Bigelow representation.

Infinite generation of the kernels of the Magnus and Burau representations

Consider the kernel Mag_g of the Magnus representation of the Torelli group and the kernel Bur_n of the Burau representation of the braid group. We prove that for g >= 2 and for n >= 6 the groups Mag_g and Bur_n have infinite rank first homology. As a consequence we conclude that neither group has any finite generating set. The method of proof in each case consists of producing a kind of "Johnson-type" homomorphism to an infinite rank abelian group, and proving the image has infinite rank. For the case of Bur_n, we do this with the assistance of a computer calculation.

Catégorification des coefficients de la matrice de la représentation de Burau

We categorify the coefficients of the Burau representation matrix using elementary geometrical methods. We show the faithfulness of this categorification in the sense that it detects the trivial braid.

Poincaré series of Klein groups, coxeter polynomials, the Burau representation, and Milnor invariants

We obtain several formulas for the Poincare series defined by B. Kostant for Klein groups (binary polyhedral groups) and some formulas for Coxeter polynomials (characteristic polynomials of monodromy in the case of singularities). Some of these formulas—the generalized Ebeling formula, the Christoffel-Darboux identity, and the combinatorial formula—are corollaries to the well-known statements on the characteristic polynomial of a graph and are analogous to formulas for orthogonal polynomials. The ratios of Poincare series and Coxeter polynomials are represented in terms of branched continued fractions, which are q-analogs of continued fractions that arise in the theory of resolution of singularities and in the Kirby calculus. Other formulas connect the ratios of some Poincare series and Coxeter polynomials with the Burau representation and Milnor invariants of string links. The results obtained by S.M. Gusein-Zade, F. Delgado, and A. Campillo allow one to consider these facts as statements on the Poincare series of the rings of functions on the singularities of curves, which suggests the following conjecture: the ratio of the Poincare series of the rings of functions for close (in the sense of adjacency or position in a series) singularities of curves is determined by the Burau representation or by the Milnor invariants of a string link, which is an intermediate object in the transformation of the knot of one singularity into the knot of the other.

Tensoring three irreducible linear representations of the braid group

The reduced Burau representation is a one-parameter representation of B n , the braid group on n strings. Specializing the parameter to a non-zero complex number x gives a representation β n (x) : B n → GL(ℂ n−1) which is either irreducible or has an irreducible composition factor . It was proved in a previous work that the tensor product of an irreducible β n (x) or with an irreducible β n (y) or is irreducible unless x = y ±1. In our work, we will consider three (irreducible) specializations of the Burau representations and we will find conditions on the non-zero complex numbers x, y and z that guarantee the irreducibility of the tensor product of the three linear representations β n (x) or , β n (y) or and β n (z) or .

On non-conjugate braids with the same closure link

We use a refinement of an argument by Shinjo, and some study of the 3-strand Burau representation, to extend from knots to links her previous construction of infinite sequences of pairwise non-conjugate braids with the same closure of a non-minimal number of (and at least 4) strands.

On the image of the Burau representation of the IA-automorphism group of a free group

In this paper we study the graded quotients of the lower central series of the image of the IA-automorphism group of a free group by the Burau representation. In particular, we determine their structures for degrees 1 and 2.

Complex Specializations of Krammer’s Representation of the Braid Group, B3

Problem statement: Classifying irreducible complex representations of an abstract group has been always a problem of interest in the field of group representations. In our study, we considered a linear representation of the braid group on three strings, namely, Krammer's representation. The objective of our work was to study the irreducibility of a specialization of Krammer's representation. Approach: We specialized the indeterminates used in defining the representation to non zero complex numbers and worked on finding invariant subspaces under certain conditions on the indeterminates. Results: we found a necessary and sufficient condition that guarantees the irreducibility of Krammer's representation of the braid group on three strings. Conclusion: This was a logical extension to previous results concerning the irreducibility of complex specializations of the Burau representation. The next step is to generalize our result for any n, which might enable us to characterize all irreducible Krammer's representations of various degrees.

Characters of the Grothendieck-Teichmüller group through rigidity of the Burau representation

We present examples of characters of absolute Galois groups of number fields that can be recovered through their action by automorphisms on the profinite completion of the braid groups, using a rigidity approach. The way we use to recover them is through classical representations of the braid groups, and in particular through the Burau representation. This enables one to extend these characters to GrothendieckTeichmuller groups.

A bound for the topological entropy of homeomorphisms of a punctured two-dimensional disk

We consider homeomorphisms ƒ of a punctured 2-disk D2 \ P, where P is a finite set of interior points of D2, which leave the boundary points fixed. Any such homeomorphism induces an automorphism ƒ* of the fundamental group of D2 \ P. Moreover, to each homeomorphism ƒ, a matrix Bƒ (t) from GL(n, ℤ[t, t−1]) can be assigned by using the well-known Burau representation. The purpose of this paper is to find a nontrivial lower bound for the topological entropy of ƒ. First, we consider the lower bound for the entropy found by R. Bowen by using the growth rate of the induced automorphism ƒ*. Then we analyze the argument of B. Kolev, who obtained a lower bound for the topological entropy by using the spectral radius of the matrix Bƒ (t), where t ∈ ℂ, and slightly improve his result.

DENSELY ORDERED BRAID SUBGROUPS

Dehornoy showed that the Artin braid groups Bn are left-orderable. This ordering is discrete, but we show that, for n &gt; 2 the Dehornoy ordering, when restricted to certain natural subgroups, becomes a dense ordering. Among subgroups which arise are the commutator subgroup and the kernel of the Burau representation (for those n for which the kernel is nontrivial). These results follow from a characterization of least positive elements of any normal subgroup of Bn which is discretely ordered by the Dehornoy ordering.