Gassner Representation Research Articles

The homomorphism defect of an extended Levine–Tristram signature via twisted homology

Taking the Levine–Tristram signature of the closure of a braid defines a map from the braid group to the integers. A formula of Gambaudo and Ghys provides an evaluation of the homomorphism defect of this map in terms of the Burau representation and the Meyer cocycle. In 2017, Cimasoni and Conway generalized this formula to the multivariable signature of the closure of colored tangles. In this paper, we extend even further their result by using a different 4-dimensional interpretation of the signature. We obtain an evaluation of the additivity defect in terms of the Maslov index and the isotropic functor [Formula: see text]. We also show that in the case of colored braids this defect can be rewritten in terms of the Meyer cocycle and the colored Gassner representation, making it a direct generalization of the formula of Gambaudo and Ghys.

A Note on the Kernel of the Gassner Representation

A Note on the Kernel of the Gassner Representation

Conway’s potential function via the Gassner representation

We show how Conway's multivariable potential function can be constructed using braids and the reduced Gassner representation. The resulting formula is a multivariable generalization of a construction, due to Kassel-Turaev, of the Alexander-Conway polynomial in terms of the Burau representation. Apart from providing an efficient method of computing the potential function, our result also removes the sign ambiguity in the current formulas which relate the multivariable Alexander polynomial to the reduced Gassner representation. We also relate the distinct definitions of this representation which have appeared in the literature.

Higher Dimensional Irreducible Representations of the Pure Braid Group

The reduced Gassner representation is a multi-parameter representation of $%P_{n},$ the pure braid group on n strings. Specializing the parameters $%t_{1},t_{2},...,t_{n}$ to nonzero complex numbers $x_{1},x_{2},...,x_{n}$gives a representation $G_{n}(x_{1},\ldots ,x_{n}):P_{n}\rightarrowGL(\mathbb{C}^{n-1})$ which is irreducible if and only if $x_{1}\ldotsx_{n}\neq 1$. In a previous work, we found a sufficient condition for theirreducibility of the tensor product of two irreducible Gassnerrepresentations. In our current work, we find a sufficient condition that guarantees the irreducibility of the tensor product of three Gassner representations. Next, a generalization of our result is given by considering the irreducibility of the tensor product of $k$ representations (\;$k \geq 3\;$).

On the irreducibility of linear representations of the pure braid group

Following up on our result in [1], we find a milder sufficient condition for the tensor product of specializations of the reduced Gassner representation of the pure braid group to be irreducible. We prove that $G_n(x_1, \ldots, x_n) \otimes G_n(y_1, \ldots, y_n) : P_n \to GL(\mathbb{C}^{n-1} \otimes \mathbb{C}^{n-1})$ is irreducible if $x_i \neq \pm y_i $ and $x_j \neq \pm {{y_j}^{-1}} $ for some $i$ and $j$.

GENERALIZED LONG-MOODY REPRESENTATIONS OF BRAID GROUPS

Long and Moody give a method of constructing representations of the braid groups Bn. We discuss some ways to generalize their construction. One of these gives representations of subgroups of Bn, including the Gassner representation of the pure braid group as a special case. Another gives representations of the Hecke algebra.

The Magnus representation and higher-order Alexander invariants for homology cobordisms of surfaces

The set of homology cobordisms from a surface to itself with markings of their boundaries has a natural monoid structure. To investigate the structure of this monoid, we define and study its Magnus representation and Reidemeister torsion invariants by generalizing Kirk-Livingston-Wang's argument over the Gassner representation of string links. Moreover, by applying Cochran and Harvey's framework of higher-order (non-commutative) Alexander invariants to them, we extract several pieces of information about the monoid and related objects.

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.

EXTENDING REPRESENTATIONS OF BRAID GROUPS TO THE AUTOMORPHISM GROUPS OF FREE GROUPS

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.

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.

PURE BRAIDS AS AUTOMORPHISMS OF FREE GROUPS

We study the composition of F. R. Cohen's map Pn → Pnk with the Gassner representation, where Pn is the pure braid group. This gives us a linear representation of Pn whose composition factors are one copy of the Gassner representation of Pn and k - 1 copies of a diagonal representation, hence a direct sum of one-dimensional representations.

WADA'S REPRESENTATIONS OF THE PURE BRAID GROUP

We consider the Magnus representation of the image of the pure braid group under the generalizations of the standard Artin representation, discovered by M. Wada. We will give a necessary and sufficient condition for the specialization of the reduced Wada's representation Gn(z) : Pn → GLn-1(ℂ) to be irreducible. It will be shown that for z = (z1,…,zn) ∈ (ℂ*)n, Gn(z) is irreducible if and only if z1k⋯znk ≠ 1. This is a generalization of our previous result concerning the irreducibility of the complex specialization of the reduced Gassner representation of Pn.

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.

Quantization of Bending Deformations of Polygons In , Hypergeometric Integrals and the Gassner Representation

AbstractThe Hamiltonian potentials of the bending deformations of n-gons instudied in [KM] and [Kly] give rise to a Hamiltonian action of the Malcev Lie algebra𝓟nof the pure braid groupPnon the moduli spaceMrofn-gon linkages with the side-lengthsr = (r1, … , rn)in. Ife∈Mris a singular point wemay linearize the vector fields in𝓟nate. This linearization yields a flat connection ∇ on the spaceof n distinct points on. We show that the monodromy of ∇ is the dual of a quotient of a specialized reduced Gassner representation.

THE GASSNER REPRESENTATION FOR STRING LINKS

The Gassner representation of the pure braid group to $GL_n(Z[Z^n])$ can be extended to give a representation of the concordance group of $n$-strand string links to $GL_n(F)$, where $F$ is the field of quotients of $\zz[\zz^n]$, $ F = Q(t_1,...,t_n)$; this was first observed by Le Dimet. Here we present simple new perspectives on its definition, its computation, and its applications to obtain both general results and interesting family of examples.

The Braid Group and Linear Rigidity

We introduce the BC-operation (short for braid-companion) on tuples of matrices. It corresponds to Katz's middle convolution operation on local systems, and generalizes it to fields K of arbitrary characteristic. It is based on the Burau–Gassner representation of the braid group. We expect many applications, in Galois theory (for finite K) as well as in Katz's original set-up of local systems and linear differential equations (where K = ℂ).

Alexander invariants of complex hyperplane arrangements

Let $\mathcal {A}$ be an arrangement of $n$ complex hyperplanes. The fundamental group of the complement of $\mathcal {A}$ is determined by a braid monodromy homomorphism, $\alpha :F_{s}\to P_{n}$. Using the Gassner representation of the pure braid group, we find an explicit presentation for the Alexander invariant of $\mathcal {A}$. From this presentation, we obtain combinatorial lower bounds for the ranks of the Chen groups of $\mathcal {A}$. We also provide a combinatorial criterion for when these lower bounds are attained.

Fixed points of endomorphisms of a free metabelian group

We consider IA-endomorphisms (i.e. Identical in Abelianization) of a free metabelian group of nite rank, and give a matrix characterization of their xed points which is similar to (yet different from) the well-known characterization of eigenvectors of a linear operator in a vector space. We then use our matrix characterization to elaborate several properties of the xed point groups of metabelian endomorphisms. In particular, we show that the rank of the xed point group of an IA-endomorphism of the free metabelian group of rank n> 2 can be either equal to 0, 1, or greater than (n 1) (in particular, it can be innite). We also point out a connection between these properties of metabelian IA-endomorphisms and some properties of the Gassner representation of pure braid groups.

The reduced Gassner representation restricted to a normal free subgroup of the pure braid group

We show that the reduced Gassner representation restricted to a normal free subgroup U n of the pure braid group, P n , is essentially the Magnus representation of U n / U '' n provided that the indeterminate y, used in defining this representation, is set to be one. Then, as a corollary, Lipschutz's result follows easily that is, in general, the kernel of the reduced Gassner representation restricted to U n is a subgroup of U '' n .

A faithfulness criterion for the Gassner representation of the pure braid group

This work is directed towards the open question of the faithfulness of the reduced Gassner representation of the pure braid group, $P_{n}(n > 3)$. Long and Paton proved that if a Burau matrix $M$ has ones on the diagonal and zeros below the diagonal then $M$ is the identity matrix. In this paper, a generalization of Long and Paton’s result will be proved. Our main theorem is that if the trace of the image of an element of $P_{n}$ under the reduced Gassner representation is $n-1$, then this element lies in the kernel of this representation. Then, as a corollary, we prove that an analogue of the main theorem holds true for the Burau representation of the braid group.