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In 1936 W. Burau discovered an interesting family of n×n matrices that give a linear representation of Artin’s classical braid group Bn, n=1,2,…. A natural question followed very quickly: is the so-called Burau representation faithful? Over the years it was proved to be faithful for n≤3, nonfaithful for n≥5, but the case of n=4 remains open to this day, in spite of many papers on the topic. This paper introduces braid groups, describes the problem in ways that make it accessible to readers with a minimal background, reviews the literature, and makes a contribution that reinforces conjectures that the Burau representation of B4 is faithful.

We show that most of the genus-zero subgroups of the braid group $\mathbb{B}_3$ (which are roughly the braid monodromy groups of the trigonal curves on the Hirzebruch surfaces) are irrelevant as far as the Alexander invariant is concerned: there is a very restricted class of \enquote{primitive} genus-zero subgroups such that these subgroups and their genus-zero intersections determine all the Alexander invariants. Then, we classify the primitive subgroups in a special subclass. This result implies the known classification of the dihedral covers of irreducible trigonal curves., Comment: 21 pages, 1 Table

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.

The Burau representation of the braid group can be used to recover the Alexander polynomial of the closure of a braid. We define twisted Burau maps and use them to compute twisted Alexander polynomials., 20 pages, 3 figures; v2: minor changes, to appear in Proc. Edinburgh Math. Soc

We consider the graph $E_{n+1,1}$ with (n+1) generators $\sigma_1,..., \sigma_{n}$, and $\delta$, where $\sigma_{i}$ has an edge with $\sigma_{i+1}$ for $i=1,...,n+1$, and $ \sigma_{1}$ has an edge with $\delta$. We then define the Artin group of the graph $E_{n+1,1}$ for $n=3$ and $n=4$ and consider its reduced Perron's representation of degrees four and five respectively. After we specialize the indeterminates used in defining the representation to non-zero complex numbers, we obtain necessary and sufficient conditions that guarantee the irreducibility of the representations for $n=3$ and $4$ .

We construct a weak 2-functor from the bicategory of oriented tangles to a bicategory of Lagrangian cospans. This functor simultaneously extends the Burau representation of the braid groups, its generalization to tangles due to Turaev and the first-named author, and the Alexander module of 1 and 2-dimensional links.

A long-standing open problem is to determine for which values of n the Burau representation Psi_n of the braid group B_n is faithful. Following work of Moody, Long-Paton, and Bigelow, the remaining open case is n = 4. One criterion states that Psi_n is unfaithful if and only if there exists a pair of arcs in the n-punctured disk D_n such that a certain associated polynomial is zero. In this paper, we use a computer search to show that there is no such arc-pair in D_4 with 2000 or fewer intersections, thus certifying the faithfulness of Psi_4 up to this point. We also investigate the structure of the set of arc-pair polynomials, observing a striking periodicity that holds between those that are, in some sense, 'closest' to zero. This is the first instance known to the authors of a deeper analysis of this polynomial set.

The (reduced) Burau representation V n of the braid group B n is obtained from the action of B n on the homology of an infinite cyclic cover of the disc with n punctures. In this paper, we calculate H *(B n ; V n ). Our topological calculation has the following arithmetic interpretation (which also has different algebraic proofs): the expected number of points on a random superelliptic curve of a fixed genus over F q is q.

We consider the linear representations of the mapping class group of an n-punctured 2-sphere constructed by V. F. R. Jones using Iwahori-Hecke algebras of type A. We show that their faithfulness is equivalent to that of certain related Iwahori-Hecke algebra representation of Artin's braid group of n-1 strands. In the case of n=6, we provide a further restriction for the kernel using our previous result, as well as a certain relation to the Burau representation of degree 4., 9 pages

We present a reduced Burau-like representation for the mixed braid group on one strand representing links in lens spaces and show how to calculate the Alexander polynomial of a link directly from the mixed braid. Vpeljemo reducirano Burauovo reprezentacijo za mešano grupo kit na eni niti, ki predstavlja splet v lečastem prostoru in pokažemo kako izračunati Alexandrov polinom spleta neposredno iz mešane kite.

For some infinite-dimensional groups $G$ and suitable subgroups $K$ there exists a monoid structure on the set $K\backslash G/K$ of double cosets of $G$ with respect to $K$. In this paper we show that the group $B_\infty$, of the braids with finitely many crossings on infinitely many strands, admits such a structure.

We give a formula of the colored Alexander invariant in terms of the homological representation of the braid groups which we call truncated Lawrence's representation. This formula generalizes the famous Burau representation formula of the Alexander polynomial., 13 pages, 3 Figures

We consider the spin representation of Artin's braid group, which has a negative index of one and was originally given by D. D. Long and explicitly computed by J.P.Tian. In our work, we find sufficient conditions under which the complex specialization of that representation, namely $\alpha :B_{n}\to GL_{n^{2}}(\mathbb C)$, is unitary relative to a nonsingular hermitian matrix.

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.

We consider the multi-parameter representation of Artin's braid group introduced by D. D. Long and J. P. Tian, namely $\alpha: B_{n}\rightarrow GL_{m}(C)$, where $m=n!n$ . First, we show that there exists a complex specialization of the multi-parameter representation that does not arise from any Hecke algebra. Second, we find conditions under which the images of the generators of the braid group on three strings under the multi-parameter representation are unitary relative to a nonsingular hermitian matrix.

We consider subgroups of the braid groups which are generated by $$n$$ th powers of the standard generators and prove that any infinite intersection (with even $$n$$ ) is trivial. This is motivated by some conjectures of Squier concerning the kernels of Burau’s representations of the braid groups at roots of unity. Furthermore, we show that the image of the braid group on 3 strands by these representations is either a finite group, for a few roots of unity, or a finite extension of a triangle group, by using geometric methods.

We describe a collection of graded rings which surject onto Webster rings for sl(2) and which should be related to certain categories of singular Soergel bimodules. In the first non-trivial case, we construct a categorical braid group action which categorifies the Burau representation.

We use some Lie group theory and the unitarizations of the Burau and Lawrence–Krammer representation, to prove that for generic parameters of definite form the image of these representations (also on certain types of subgroups) is dense in the unitary group. This implies that, except possibly for closures of full-twist braids, all links have infinitely many conjugacy classes of braid representations on any non-minimal number of (and at least 4) strands.

The problem of faithfulness of the (reduced) Burau representation for [Formula: see text] is known to be equivalent to the problem of whether certain two matrices [Formula: see text] and [Formula: see text] generate a free group of rank two. In this note, we give a simple proof that [Formula: see text] is a free group of rank two.

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 .

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.

A group-theoretical method, via Wada's representations, is presented to distinguish Kishino's virtual knot from the unknot. Biquandles are constructed for any group using Wada's braid group representations. Cocycle invariants for these biquandles are studied. These invariants are applied to show the non-existence of Alexander numberings and checkerboard colorings., Comment: 14 pages, 8 figures

The reduced Gassner representation is a multi-parameter representation of Pn, the pure braid group on n strings. Specializing the parameters t1, t2,...,tn to nonzero complex numbers x1,x2,...,xn gives a representation Gn(x1,...,xn): Pn GL( n 1 ) which is irreducible if and only if x1...xn 1.We find a sufficient condition that guarantees that the tensor product of an irreducible Gn(x1,...,xn)with an irreducible Gn(y1, ..., yn) is irreducible. We fall short of finding a necessary and sufficient condition for irreducibility of the tensor product. Our work is a continuation of a previous one regarding the tensor product of complex specializations of the Burau representation of the braid group.

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.

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.

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.

The topological entropy of a braid is the infimum of the entropies of all homeomorphisms of the disc which have a finite invariant set represented by the braid. When the isotopy class represented by the braid is pseudo-Anosov or is reducible with a pseudo-Anosov component, this entropy is positive. Fried and Kolev proved that the entropy is bounded below by the logarithm of the spectral radius of the braid's Burau matrix, $B(t)$, after substituting a complex number of modulus~1 in place of $t$. In this paper we show that for a pseudo-Anosov braid the estimate is sharp for the substitution of a root of unity if and only if it is sharp for $t=-1$. Further, this happens if and only if the invariant foliations of the pseudo-Anosov map have odd order singularities at the strings of the braid and all interior singularities have even order. An analogous theorem for reducible braids is also proved., Comment: 28 pages, 8 figures

We consider the Magnus representation of the image of the braid group under a representation discovered by Wada. We first investigate the question about the reducibility of this representation and show that it is of Burau type. Next, we show that the images of the generators of the braid group under that representation are unitary relative to a hermitian matrix. This is similar to the well known result by Squier that asserts that the Burau representation of the braid group is unitary.

On complex linear spaces, Fuchs-type Pfaffian systems are studied that are defined by configurations of vectors in these spaces. These systems are referred to as R-systems in this paper. For the vector configurations that are systems of roots of complex reflection groups, the monodromy representations of R-systems are described. These representations are deformations of the standard representations of reflection groups. Such deformations define representations of generalized braid groups corresponding to complex reflection groups and are similar to the Burau representations of the Artin braid groups.

This paper establishes that sutured annular Khovanov homology is not invariant for braid closures under axis-preserving mutations. This follows from an explicit relationship between sutured annular Khovanov homology and the classical Burau representation for braid closures., Comment: 14 pages, 7 figures

In this paper, we give a survey of recent results devoted to the homology of generalizations of braids: the homological properties of virtual braids and the generalized homology of Artin groups studied by C. Broto and the author. Virtual braid groups VBn correspond to virtual knots in the same way that classical braids correspond to usual knots. Virtual knots arise in the study of Gauss diagrams and Vassiliev invariants of usual knots. The Burau representation to GLnℤ[t, t−1] is extended from classical braids to virtual ones. Its homological properties are also studied. The following splitting of infinite loop spaces for the plus-construction of the classifying space of the virtual braid group on an infinite number of strings exists: $$\mathbb{Z} \times BV{\kern 1pt} B_\infty ^ + \simeq \Omega ^\infty S^\infty \times S^1 \times Y,$$ where Y is an infinite loop space. Connections with K*ℤ are discussed. In the last section, information on Morava K-theory and the Brown-Peterson homology of Artin groups and braid groups in handlebodies is collected.

In this paper, it is proved that the intersection of the radicals of nilpotent residues for the generalized pure braid group corresponding to an irreducible finite Coxeter group or an irreducible imprimitive finite complex reflection group is always trivial. The proof uses the solvability of the Riemann—Hilbert problem for analytic families of faithful linear representations by the Lappo-Danilevskii method. Generalized Burau representations are defined for the generalized braid groups corresponding to finite complex reflection groups whose Dynkin—Cohen graphs are trees. The Fuchsian connections for which the monodromy representations are equivalent to the restrictions of generalized Burau representations to pure braid groups are described. The question about the faithfulness of generalized Burau representations and their restrictions to pure braid groups is posed.

We construct a functor from the category of oriented tangles in R 3 to the category of Hermitian modules and Lagrangian relations over Z[t, t −1 ]. This functor extends the Burau representations of the braid groups and its generalization to string links due to Le Dimet. 2005 Elsevier Ltd. All rights reserved.

The kernel of Burau(4) Zp, the reduced Burau representation with coecients in Zp of the 4-braid group B4, consists only of pseudo-Anosov braids.

We extend the Burau representation to the group Cn of conjugating automorphisms. We extend the Lawrence—Krammer faithful linear representation of the braid group B3 to C3, and for n ≥ 4 we extend this representation under certain restrictions on the parameters of the representation. We determine that the spherical braid group Bn(S2) and the mapping class group M(0, n) of an n-punctured sphere are linear for all n ≥ 2. The automorphism group Aut(Fn) is not linear for n ≥ 3, and the group Aut(F2) is linear iff so is the braid group B4. Using the Lawrence—Krammer representation, we construct a faithful linear representation of Aut(F2).

We consider the Magnus representation of the image of the braid group under the generalizations of the standard Artin representation discovered by M. Wada. We show that the images of the generators of the braid group under the Magnus representation are unitary relative to a Hermitian matrix. As a special case, we get that the Burau representation is unitary, which was known and proved by C. C. Squier.

In this Note, we define infinitesimal analogues of the Iwahori–Hecke algebras associated with finite Coxeter groups. These are reductive Lie algebras for which we announce several decomposition results. These decompositions yield irreducibility results for representations of the corresponding (pure) generalized braid groups deduced from Hecke algebra representations through tensor constructions. To cite this article: I. Marin, C. R. Acad. Sci. Paris, Ser. I 337 (2003).

A lot of linear representations of the braid group, Bn, arise as a result of treating braids as automorphisms of a free group. In this paper, we consider the composition of F. R. Cohen's map Bn → Bnk and the embedding Bnk → Aut (Fnk). This gives us a linear representation of Bn whose composition factors are one copy of the Burau representation and k - 1 copies of the standard representation, a representation investigated by I. Sysoeva.

Nous definissons et entamons l'etude d'analogues infinitesimaux des quotients principaux (algebres de Temperley-Lieb, Hecke, Birman-Wenzl-Murakami) de l'algebre de groupe du groupe d'Artin B n . Ce sont des algebres de Hopf qui correspondent a des groupes reductifs, et permettent de donner un cadre general aux representations derivees des representations classiques de B n . Nous decomposons completement l'algebre de Temperley-Lieb infinitesimale, et en deduisons plusieurs resultats d'irreductibilite.

A special class of integrable Fuchsian systems on Cn related to KZ equations is considered. We survey the construction of such systems and the list of the structural properties their monodromy representations. The relation of the Fuchsian systems obtained by the Veselov construction assosiated with a deformation of the An−1-type root system and the Gauss–Manin connection of the natural projection Cn→Cn−1 is described. In this case, we prove that the monodromy representation is equivalent to the Burau representation of the Artin braid group. For a deformations of the other root system, we introduce generalized Burau representations. We conjecture that the integrable Fuchsian systems related to essential new finite sets of the vectors described by Veselov and Chalykh are the result of the Klares–Schlesinger isomonodromic deformations (or transformation) of the integrable Fuchsian system related to the Coxeter root systems.

Let n ≥ 2 be an integer and ℬn the n-braid group. A braid β ∈ ℬn is said to be a (p,s)-lens braid if there exists α ∈ ℬn such that β = αp(σ1 σ2 … σn-1)ns. In this paper we use the multi-variable Alexander polynomial to find a necessary condition for a braid to be a (p,s)-lens braid, for p prime. Our main tool here is the multi-variable Burau representation of the n-braid group.

Pseudo-Anosov homeomorphisms are classified by their invariant train tracks. The decomposition of any train track map in elementary folding maps gives a normal form for each train track class. In the case of 4-braids there are three train tracks classes and we give an explicit automaton that generates a normal form for each class. This enables us, for instance, to exhibit the pseudo-Anosov 4-braid with the minimal growth rate. We also show that the growth rate of a pseudo-Anosov braid appears as a root of the Alexander polynomial of a link that shares a common sub-link with the closure of the braid. We finally give a criterion for the faithfulness of the Burau representation for 4-braids.

We address the question: Does there exist a non-trivial knot with a trivial Jones polynomial? To find such a knot, it is almost certainly sufficient to find a non-trivial braid on four strands in the kernel of the Burau representation. I will describe a computer algorithm to search for such a braid.

We establish relations between both the classical and the dual Garside structures of the braid group and the Burau representation. Using the classical structure, we formulate a non-vanishing criterion for the Burau representation of the 4-strand braid group. In the dual context, it is shown that the Burau representation for arbitrary braid index is injective when restricted to the set of simply-nested braids.

It is well known that the Artin representation of the braid group may be used to calculate the fundamental group of knot complements while the Burau representation can be used to calculate the Alexander polynomial of knots. In this note we will study extensions of the Artin representation and the Burau representation to the singular braid monoid and the relation between them which are induced by Fox' free calculus. Closing singular braids, we obtain singular knots as they appear in the theory of Vassiliev invariants. Thus the extensions of the Artin representation and the Burau presentation give rise to invariants of singular knots. Here, we will focus on the invariants coming from the extended Artin representations. We obtain an infinite family of group invariants, all of them in relation with the fundamental group of the knot complement.

A 2-variable matrix B ∈ GLn(Z[u±1, v±1]) is defined for any n-string link, generalizing the Burau matrix of an nbraid. The specialization u = 1, v = t−1 recovers the generalized Burau matrix recently defined by X. S. Lin, F. Tian and Z. Wang using probabilistic methods. The specialization u = t−1, v = 1 results in a matrix with a natural algebraic interpretation, and one that yields homological information about the complement of the closed string link.

The braid group B n B_n can be defined as the mapping class group of the n n -punctured disk. A group is said to be linear if it admits a faithful representation into a group of matrices over R \mathbf R . Recently Daan Krammer has shown that a certain representation of the braid groups is faithful for the case n = 4 n=4 . In this paper, we show that it is faithful for all n n .

Denote by Φ2 the automorph ism group of the free group F2 on two generators. We classify the irreducible 6-dimensional complex representations of Φ2 whose restriction to F2 is nontrivial. J. Dyer, E. Formanek, and E. Grossman have shown how the Burau representation of the braid group B2 gives rise to a oneparameter famil y of irreducible 6-dimensional representations of Φ2. The faithfulness question for these and some other clo sely related representations of Φ2 is open. Our classification shows that all other 6-dimensional representations of Φ2 are not faithful.

Using a probabilistic interpretation of the Burau representation of the braid group oered by Vaughan Jones, we generalize the Burau representation to a representation of the semigroup of string links. This representation is determined by a linear system, and is dominated by nite type string link invariants. For positive string links, the representation matrix can be interpreted as the transition matrix of a Markov process. For positive non-separable links, we show that all states are persistent.

This work is directed towards the open question of the faithfulness of the reduced Gassner representation of the pure braid group, Pn(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 Pn 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.