Classical Braid Group Research Articles

Cohomology of affine Artin groups and applications

The result of this paper is the determination of the cohomology of Artin groups of type An, B n and A n with non-trivial local coefficients. The main result is an explicit computation of the cohomology of the Artin group of type B n with coefficients over the module Q[q ±1 ,t ±1 ]. Here the first n - 1 standard generators of the group act by (-q)-multiplication, while the last one acts by (-t)-multiplication. The proof uses some technical results from previous papers plus computations over a suitable spectral sequence. The remaining cases follow from an application of Shapiro's lemma, by considering some well-known inclusions: we obtain the rational cohomology of the Artin group of affine type An as well as the cohomology of the classical braid group Br n with coefficients in the n-dimensional representation presented in Tong, Yang, and Ma (1996). The topological counterpart is the explicit construction of finite CW-complexes endowed with a free action of the Artin groups, which are known to be K(π, 1) spaces in some cases (including finite type groups). Particularly simple formulas for the Euler-characteristic of these orbit spaces are derived.

The homology of the Milnor fiber for classical braid groups

In this paper we compute the homology of the braid groups, with coefficients in the module Z[q^+-1] given by the ring of Laurent polynomials with integer coefficients and where the action of the braid group is defined by mapping each generator of the standard presentation to multiplication by -q. The homology thus computed is isomorphic to the homology with constant coefficients of the Milnor fiber of the discriminantal singularity.

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.

Algebraic Markov equivalence for links in three-manifolds

Let B-n denote the classical braid group on n strands and let the mixed braid group B-m,B-n be the subgroup of Bm+n comprising braids for which the first m strands form the identity braid. Let B-m,B-infinity = boolean OR(n) B-m,B-n. We describe explicit algebraic moves on B-m,B-infinity such that equivalence classes under these moves classify oriented links up to isotopy in a link complement or in a closed, connected, oriented three-manifold. The moves depend on a fixed link representing the manifold in S-3. More precisely, for link complements the moves are the two familiar moves of the classical Markov equivalence together with 'twisted' conjugation by certain loops a(i). This means premultiplication by a(i)(-1) and postmultiplication by a 'combed' version of a(i). For closed three-manifolds there is an additional set of 'combed' band moves that correspond to sliding moves over the surgery link. The main tool in the proofs is the one-move Markov theorem using L-moves (adding in-box crossings). The resulting algebraic classification is a direct extension of the classical Markov theorem that classifies links in S-3 up to isotopy, and potentially leads to powerful new link invariants, which have been explored in special cases by the first author. It also provides a controlled range of isotopy moves, useful for studying skein modules of three-manifolds.

Parametrized braid groups of Chevalley groups.

We introduce the notion of a braid group parametrized by a ring, which is defined by generators and relations and based on the geometric idea of painted braids. We show that the parametrized braid group is isomorphic to the semi-direct product of the Steinberg group (of the ring) with the classical braid group. The technical heart of the proof is the Pure Braid Lemma, which asserts that certain elements of the parametrized braid group commute with the pure braid group. This first part treats the case of the root system A_n ; in the second part we prove a similar theorem for the root system D_n .

The virtual and universal braids

We study the structure of the virtual braid group. It is shown that the virtual braid group is a semi-direct product of the virtual pure braid group and the symmetric group. Also, it is shown that the virtual pure braid group is a semi-direct product of free groups. From these results we obtain a normal form of words in the virtual braid group. We introduce the concept of a universal braid group. This group contains the classical braid group and has as its quotient groups the singular braid group, virtual braid group, welded braid group, and classical braid group. Recently some generalizations of classical knots and links were de- fined and studied: singular links (20, 5), virtual links (15, 12) and welded links (10). One of the ways to study classical links is to study the braid group. Singular braids (1, 5), virtual braids (15, 21), welded braids (10) were defined similar to the classical braid group. A theorem of A. A. Markov (4, Ch. 2.2) reduces the problem of classification of links to some alge- braic problems of the theory of braid groups. These problems include the word problem and the conjugacy problem. There are generaliza- tions of Markov's theorem for singular links (11), virtual links, and welded links (14). There are some dierent ways to solve the word problem for the singular braid monoid and singular braid group (8, 7, 22). The solution of the word problem for the welded braid group follows from the fact that this group is a subgroup of the automorphism group of the free group (10). A normal form of words in the welded braid group was constructed in (13). In this paper we study the structure of the virtual braid group V Bn. This is similar to the classical braid group Bn and welded braid group

On Generalized Homology of Artin Groups

The Morava K-theory and the generalized Brown―Peterson homology of groups related to the classical braid groups are calculated. Examples of such groups are the Artin groups and braid groups in handlebodies. Bibliography: 13 titles.

ORIENTED DIVIDES AND PLANE CURVE BRAIDS

This paper begins with a review of the extension of A'Campo's divides to the more general construction of oriented divides, using the same disk model of S3. It is shown that all links in S3 can be represented by oriented divides (in contrast with divide links) and simple methods are presented for moving between links and their oriented divide presentations. Using the properties of the disk model, oriented divides are extended by three extra equivalence rules, and it is shown that this extended theory of oriented divides is equivalent to knot theory in S3. Fueled by this result, the concept of oriented divide braid is introduced, together with a group construction which parallels the classical braid group representation of geometric braids. This leads to results analagous to Alexander's well-known theorem concerning braid presentations, and also the Markov theorem.

Vector braids

In this paper we define a new family of groups which generalize the classical braid groups on C . We denote this family by { B m n } n ≥ m + 1 where n, m ϵ N . The family {B 1 n} n ϵ N is the set of classical braid groups on n strings. The group B m n is related to the set of motions of n unordered points in C m , so that at any time during the motion, each m + 1 of the points span the whole of C m in the sense of affine geometry. There is a map from B m n to the symmetric group on n letters. We let P m n denote the kernel of this map. In this paper we are mainly interested in finding a presentation of and understanding the group P 2 n . We give a presentation of a group PL n which maps surjectively onto P 2 n . We also show the surjection PL n → P 2 n induces an isomorphism on first and second integral homology and conjecture that it is an isomorphism. We then find an infinitesimal presentation of the group P 2 n . Finally, we also consider the analagous groups where points lie in P m instead of C m . These groups generalize of the classical braid groups on the sphere.

Braid subgroup normalisers, commensurators and induced representations

The classical braid groups of E. Artin form an ascending sequence B1 B2 . . . B1. These groups, and their representations, play an important roA le in several areas of mathematics and physics. The object of this note is to establish new algebraic information about this sequence of groups. In particular, we determine the normaliser and commensurator of the braid group Bn in Bm, n m 1. They are described in terms of the centraliser, which was recently characterised in [FRZ], and this leads to explicit generators and relations presenting these subgroups. In Sect. 5 we characterise the commensurator of Bn as a certain stabilizer, under the identi®cation of Bm as the mapping class group of the m-punctured 2-dimensional disk. This identi®cation gives rise to in®nitely many natural ``geometric'' inclusions of Bn in Bm, which we demonstrate to be mutually incommensurable, although they are all conjugate. Indeed, this point of view a€ords a simple description of the centralisers, normalisers and commensurators of all the geometric braid subgroups (Theorem 5.3). In Sect. 6 we show that the action of Bm, as the mapping class group, upon appropriate sets of curves, is a transitive and large action in the sense of Burger and de la Harpe [BH]. The paper concludes with some applications regarding unitary representations of the braid groups, and those induced by braid subgroups. Here is a summary of our principal results regarding the Bn sequence; the relevant de®nitions will follow.

The top-cohomology of Artin groups with coefficients in rank-1 local systems over [formula omitted

Let W be a Coxeter group and let G w be the associated Artin group. We consider the local system over k( G w , 1) with coefficients in R = Z[q, q −1] which associates to the standard generators of G w the multiplication by q. For the all list of finite irreducible Coxeter groups we calculate the top-cohomology of this local system. It turns out that the ideal which we compute is a sort of Alexander ideal for a hypersurface. In case of the classical braid group Br n this ideal is the principal ideal generated by the nth cyclotomic polynomial. We use these results to calculate the topological category of k( G w , 1): we prove that it equals the obvious bound given by obstruction theory (so, in case of braid group Br n , it is exactly n).

The braid-permutation group

We consider the subgroup of the automorphism group of the free group generated by the braid group and the permutation group. This is proved to be the same as the subgroup of automorphisms of permutation-conjugacy type and is represented by generalised braids (braids in which some crossings are allowed to be “welded”). As a consequence of this representation there is a finite presentation which shows the close connection with both the classical braid and permutation groups. The group is isomorphic to the automorphism group of the free quandle and closely related to the automorphism group of the free rack. These automorphism groups are connected with invariants of classical knots and links in the 3-sphere.

Marked homeomorphisms and the realization problem

The role played by the classical braid groups in the interplay between geometry, algebra and topology (see [Ca]) derives, in part, from their definition as the fundamental groups of configuration spaces of points in the plane. Seeking to generalize these groups and to understand them better, one is led to ask: are there other discrete groups whose topological invariants arise from configuration spaces?The groups of marked homeomorphisms (1·1) provide a positive response which is in some sense banal; the realization problem (1·5) is to find non-banal examples.

On generalized braid groups

Braid groups were introduced by Artin [1]. These groups have been studied extensively—see [2], [9] and the references cited there. Recently work has been done on “circular” braid groups and other “braid-like” groups [7], [10]. In this paper we formulate the concept of a generalized braid group, and we begin a study of the structure of such groups. In particular for such a group G1, there is a homomorphism from G onto the infinite cyclic group, the kernel of which is the derived group G1 of G. We study G1. Our results generalize results of Gorin and Lin [5], who considered the case when G is a classical braid group B(n ≥ 3). They showed that is free abelian of rank 2 if n = 3, 4 and is trivial if n ≥ 5. They also showed that is finitely presented.

On the Stable Equivalence of Plat Representations of Knots and Links

We are interested in the question of the decideability of the classical knot problem. A knot is the embedded image of a circle S1 in Euclidean 3-space E3. If L1, L2 are knots, then L1 ≈ L2 if there is an orientation-preserving homeomorphism h: E3 ⟶ E3 with h(L1) = L2. By the “knot problem” we mean: given two arbitrary tame knots L1, L2 (a knot is tame if it is equivalent to a polygonal knot), decide in a finite number of steps whether L1 ≈ L2. The object of this paper is to show that the knot problem is ‘'stably equivalent“ to a problem of deciding membership in the double cosets of a distinguished subgroup K2n of the classical braid group B2n [1].