Criteria of prime links in terms of pseudo-characters

Pseudo-characters of groups have recently found applications in the theory of classical knots and links in ℝ³. More precisely, there is a connection between pseudo-characters of Artin’s braid groups and properties of links represented by braids. In the present work, this connection is investigated and the notion of kernel pseudo-characters of braid groups is introduced. It is proved that a kernel pseudo-character ϕ and a braid β satisfy Ιϕ(β)І > C _ϕ, where C _ϕ is the defect of ϕ, then β represents a prime link (i.e., a link that is noncomposite, nonsplit, and nontrivial). Furthermore, the space of braid group pseudo-characters is studied and a way to obtain nontrivial kernel pseudo-characters from an arbitrary braid group pseudo-character that is not a homomorphisrn is described. This allows one to use an arbitrary nontrivial braid group pseudo-character for recognition of prime knots and links. Bibliography: 17 titles.     

Generalization of Braids and Homologies

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 VB _n 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 GL _n ℤ[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:

Braid groups in genetic code

For every genetic code with finitely many generators and at most one relation, a braid group is introduced. The construction presented includes the braid group of a plane, braid groups of closed oriented surfaces, Artin— Brieskorn braid groups of series B, and allows us to study all of these groups from a unified standpoint. We clarify how braid groups in genetic code are structured, construct words in the normal form, look at torsion, and compute width of verbal subgroups. It is also stated that the system of defining relations for a braid group in two-dimensional manifolds presented in a paper by Scott is inconsistent. Supported by RFBR grant No. 02-01-01118. __________ Translated from Algebra i Logika, Vol. 45, No. 2, pp. 131–158, March–April, 2006.     

On representations and <Emphasis Type="Italic">K</Emphasis>-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. Received: 6 February 2002 / Revised version: 19 September 2002 / Published online: 8 April 2003 RID="⋆" ID="⋆" Partially supported by the NSF RID="⋆⋆" ID="⋆⋆" Partially supported by grant LEQSF(1999-02)-RD-A-01 from the Louisiana Board of Regents, and by grant MDA904-00-1-0038 from the National Security Agency RID="⋆" ID="⋆" Partially supported by the NSF Mathematics Subject Classification (2000): 20F36, 32S22, 55N15, 55R50     

Finite quotients of the pure symplectic braid group

We show how the finite symplectic groups arise as quotients of the pure symplectic braid group. Via [SV] certain of these groups — in particular, all groups Sp _n (2) — occur as Galois groups over ℚ. Supported by NSF grant DMS-9306479.     

Embedding right-angled Artin groups into graph braid groups

We construct an embedding of any right-angled Artin group G(Δ) defined by a graph Δ into a graph braid group. The number of strands required for the braid group is equal to the chromatic number of Δ. This construction yields an example of a hyperbolic surface subgroup embedded in a two strand planar graph braid group.      

On Generalized Homology of Artin Groups

The Morava K-theory and the generalized BrownPeterson 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.     

On fundamental groups related to the Hirzebruch surface <Emphasis Type="Italic">F</Emphasis><Subscript>1</Subscript>

Given a projective surface and a generic projection to the plane, the braid monodromy factorization (and thus, the braid monodromy type) of the complement of its branch curve is one of the most important topological invariants, stable on deformations. From this factorization, one can compute the fundamental group of the complement of the branch curve, either in ℂ² or in ℂℙ². In this article, we show that these groups, for the Hirzebruch surface F _1,(a,b), are almost-solvable. That is, they are an extension of a solvable group, which strengthen the conjecture on degeneratable surfaces. This work was supported by the Emmy Noether Institute Fellowship (by the Minerva Foundation of Germany) and Israel Science Foundation (Grant No. 8008/02-3)     

Braids,mapping class groups,and categorical delooping

Dehn twists around simple closed curves in oriented surfaces satisfy the braid relations. This gives rise to a group theoretic map from the braid group to the mapping class group. We prove here that this map is trivial in homology with any trivial coefficients in degrees less than g/2. In particular this proves an old conjecture of J. Harer. The main tool is categorical delooping in the spirit of (Tillmann in Invent Math 130:257–175, 1997). By extending the homomorphism to a functor of monoidal 2-categories, is seen to induce a map of double loop spaces on the plus construction of the classifying spaces. Any such map is null-homotopic. In an appendix we show that geometrically defined homomorphisms from the braid group to the mapping class group behave similarly in stable homology. The first author was supported by Inha University research grant.     

On the group of conjugating automorphisms of a free group

The group of conjugating automorphisms of a free group and certain subgroups of this group, namely, the group of McCool basis-conjugating automorphisms and the Artin braid group are considered. The Birman theorem on the representation of a braid group by matrices is sharpened. Translated fromMatematicheskie Zametki, Vol. 60, No. 1, pp. 92–108, July, 1996.     

Maximally Clustered Elements and Schubert Varieties

We introduce and study a class of “maximally clustered” elements for simply laced Coxeter groups. Such elements include as a special case the freely braided elements of Green and the author, which in turn constitute a superset of the iji-avoiding elements of Fan. We show that any reduced expression for a maximally clustered element is short-braid equivalent to a “contracted” expression, which can be characterized in terms of certain subwords called “braid clusters”. We establish some properties of contracted reduced expressions and apply these to the study of Schubert varieties in the simply laced setting. Specifically, we give a smoothness criterion for Schubert varieties indexed by maximally clustered elements. Received December 30, 2005     

Homology and cohomology groups of the group homomorphism

As is known, the homology and cohomology Massa-Takasu groups for pairs of groups (G, H) are defined by the embedding f: H → G, H < G [2]. In our case, these definitions are extended to an arbitrary group homomorphism φ: Π → G. In particular, we define homology and cohomology groups of the nth order for the homomorphism φ, and if Π = H, we obtain the known theory [2]. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 43, Topology and Its Applications, 2006.     

Asymptotic linearity of the mapping class group and a homological version of the Nielsen–Thurston classification

We study the action of the mapping class group on the integral homology of finite covers of a topological surface. We use the homological representation of the mapping class to construct a faithful infinite-dimensional representation of the mapping class group. We show that this representation detects the Nielsen–Thurston classification of each mapping class. We then discuss some examples that occur in the theory of braid groups and develop an analogous theory for automorphisms of free groups. We close with some open problems.     

On the inverse braid and reflection monoids of type <Emphasis Type="Italic">B</Emphasis>

There are well-known relations between braid and symmetric groups as well as Artin-Brieskorn braid groups and Coxeter groups: the latter are the factor-groups of the Artin-Brieskorn braid groups. The inverse braid monoid is related to the inverse symmetric monoid in the same way. We show that similar relations exist between the inverse braid monoid of type B and the inverse reflection monoid of type B. This gives a presentation of the latter monoid.     

An acyclic extension of the braid group

We relate Artin's braid groupB _℞=lim_→B_n to a certain groupF′ ofpl-homeomorphisms of the interval. Namely, there exists a short exact sequence 1→B _∞→A→F′→1 whereH _kA=0,k≥1.     

Graph theoretic method for determining hurwitz equivalence in the symmetric group

Motivated by the problem of Hurwitz equivalence of Δ² factorization in the braid group, we address the problem of Hurwitz equivalence in the symmetric group, of 1s _n factorizations with transposition factors. Looking at the transpositions as the edges in a graph, we show that two factorizations are Hurwitz equivalent if and only if their graphs have the same weighted connected components. The graph structure allows us to compute Hurwitz equivalence in the symmetric group. Using this result, one can compute non-Hurwitz equivalence in the braid group. This paper is part of the author’s PhD thesis. This work was partially supported by the Emmy Noether Research Institute for Mathematics (center of the Minerva Foundation of Germany), the Excellency Center “Group Theoretic Methods in the Study of Algebraic Varieties” of the Israel Science Foundation, and by EAGER (EU network, HPRN-CT-2009-00099). Received December 31, 2001 and in revised form August 6, 2002     

The homology and cohomology of the complements to some arrangements of codimension two complex planes

We consider the homology and cohomology of the complement to the arrangement Z = ∪_{1<|i−j|<d−1}{z _i = z _j = 0} of coordinate planes in ℂ ^d , and explicitly construct a basis for these groups as well as a basis for the homology groups of the one-point compactification of Z.     

Markov classes in certain finite quotients of artin’s braid group

This paper studies three finite quotients of the sequence of braid groups {B _n;n = 1,2,…}. Each has the property that Markov classes in {ie160-1} = ∐B _n pass to well-defined equivalence classes in the quotient. We are able to solve the Markov problem in two of the quotients, obtaining canonical representatives for Markov classes and giving a procedure for reducing an arbitrary representative to the canonical one. The results are interpreted geometrically, and related to link invariants of the associated links and the value of the Jones polynomial on the corresponding classes. This material is based upon work partially supported by the National Science Foundation under Grant No. DMS-8503758.