Intersecting subgroups and symmetric commutator subgroups in virtual link groups

Giving a presentation of the group of a 2-braid virtual knot or link, we consider the groups of three families of 2-braid virtual knots. Each of them has a certain feature; for example, we can show: for any positive integer N, there exists a virtual knot group with an element of order N. It is known that the collection of the virtual knot groups is the same as that of the ribbon T2-knot groups. Using our examples we discuss the relationship among the virtual knot groups and other knot groups such as ribbon S2-knot groups, S2-knot groups, T2-knot groups, and S3-knot groups.

Given a virtual knot K, we introduce a new group-valued invariant VGK called the virtual knot group, and we use the elementary ideals of VGK to define invariants of K called the virtual Alexander invariants. For instance, associated to the zeroth ideal is a polynomial HK(s, t, q) in three variables which we call the virtual Alexander polynomial, and we show that it is closely related to the generalized Alexander polynomial GK(s, t) introduced by Sawollek; Kauffman and Radford; and Silver and Williams. We define a natural normalization of the virtual Alexander polynomial and show it satisfies a skein formula. We also introduce the twisted virtual Alexander polynomial associated to a virtual knot K and a representation ϱ : VGK → GLn(R), and we define a normalization of the twisted virtual Alexander polynomial. As applications we derive bounds on the virtual crossing numbers of virtual knots from the virtual Alexander polynomial and twisted virtual Alexander polynomial.

Study of stable isotopy classes of a finite collection of immersed circles without triple or higher intersections on closed oriented surfaces is considered as a planar analogue of virtual knot theory, a far reaching generalisation of classical knot theory. Recent works have established Alexander and Markov theorems in the planar setting. In the classical case, the role of groups is played by twin groups, a class of right-angled Coxeter groups. A new class of groups called virtual twin groups, that extends twin groups in a natural way, plays the role of groups in the virtual case. The virtual twin group V Tn contains the pure virtual twin group P V Tn, a planar analogue of the pure Artin braid group. In this paper, we prove that the pure virtual twin group P V Tn is an irreducible right-angled Artin group with trivial center and give it's precise presentation. We show that P V Tn has a decomposition as an iterated semi-direct product of infinite rank free groups. We give a complete description of the automorphism group of P V Tn and establish splitting of natural exact sequences of automorphism groups. As applications, we show that V Tn is residually finite and P V Tn has the R∞-property.

Virtual knots undetected by 1- and 2-strand bracket polynomials

foreword by Ronald Rice The vast, international web of computer networks that is the Internet offers millions of users the opportunity to exchange electronic mail, photographs, and sound clips; to search databases for books, CDs, cars, and term papers; to participate in real-time audio- and video-conferencing; and to shop for products both virtual and physical. This huge conglomerate of links, hyperlinks, and virtual links is not just a technology for linking computers--it is a medium for communication.The convergence of computer and communication technologies creates a social convergence as well. People meet in chat rooms and discussion groups to converse on everything from auto mechanics to postmodern art. Networked groups form virtually and on-the-fly, as common interests dictate. Like interpersonal communication, the networks are participatory, their content made up by their audience. Like mass-mediated communication, they involve large audiences. But the networks are neither purely interpersonal nor purely mass--they are a new phenomenon.Network and Netplay addresses the mutual influences between information technology and group formation and development, to assess the impact of computer-mediated communications on both work and play. Areas discussed include the growth and features of the Internet, network norms and experiences, and the essential nature of network communication.Contributors : Michael Berthold, Lee Li-Jen Chen, Richard Coyne, Brenda Danet, Patrick Doyle, Brian R. Gaines, Barbara Hayes-Roth, Steve Jones, Sandra Katzman, Edward Mabry, Richard MacKinnon, Margaret McLaughlin, Sid Newton, Kerry Osborne, Sheizaf Rafaeli, Yehudit Rosenbaum-Tamari, Lucia Ruedenberg, Christine Smith, Fay Sudweeks, Alexander Voiskounsky, Diane Witmer.

Non-abelian tensor product and circular orderability of groups

Let [Formula: see text] be the algebra of Laurent polynomials in the variable [Formula: see text] and let [Formula: see text] be the algebra of Laurent polynomials in the variable [Formula: see text] and standard polynomials in the variables [Formula: see text] For [Formula: see text] we denote by [Formula: see text] the virtual braid group on [Formula: see text] strands. We define two towers of algebras [Formula: see text] and [Formula: see text] in terms of diagrams. For each [Formula: see text] we determine presentations for both, [Formula: see text] and [Formula: see text]. We determine sequences of homomorphisms [Formula: see text] and [Formula: see text], we determine Markov traces [Formula: see text] and [Formula: see text], and we show that the invariants for virtual links obtained from these Markov traces are the [Formula: see text]-polynomial for the first trace and the arrow polynomial for the second trace. We show that, for each [Formula: see text] the standard Temperley–Lieb algebra [Formula: see text] embeds into both, [Formula: see text] and [Formula: see text], and that the restrictions to [Formula: see text] of the two Markov traces coincide.

Two virtual link diagrams are homotopic if one may be transformed into the other by a sequence of virtual Reidemeister moves, classical Reidemeister moves, and self crossing changes. We recall the pure virtual braid group. We then describe the set of pure virtual braids that are homotopic to the identity braid.

The singular braids with [Formula: see text] strands, [Formula: see text], were introduced independently by Baez and Birman. It is known that the monoid formed by the singular braids is embedded in a group that is known as singular braid group, denoted by [Formula: see text]. There has been another generalization of braid groups, denoted by [Formula: see text], [Formula: see text], which was introduced by Fang as a group of symmetries behind quantum quasi-shuffle structures. The group [Formula: see text] simultaneously generalizes the classical braid group, as well as the virtual braid group on [Formula: see text] strands. We investigate the commutator subgroups [Formula: see text] and [Formula: see text] of these generalized braid groups. We prove that [Formula: see text] is finitely generated if and only if [Formula: see text], and [Formula: see text] is finitely generated if and only if [Formula: see text]. Further, we show that both [Formula: see text] and [Formula: see text] are perfect if and only if [Formula: see text].

Let [Formula: see text]. In this paper, we show that for any abelian subgroup [Formula: see text] of [Formula: see text] the crystallographic group [Formula: see text] has Bieberbach subgroups [Formula: see text] with holonomy group [Formula: see text]. Using this approach, we obtain an explicit description of the holonomy representation of the Bieberbach group [Formula: see text]. As an application, when the holonomy group is cyclic of odd order, we study the holonomy representation of [Formula: see text] and determine the existence of Anosov diffeomorphisms and Kähler geometry of the flat manifold [Formula: see text] with fundamental group the Bieberbach group [Formula: see text].

On cohomology of crystallographic groups with cyclic holonomy of split type

The big fundamental group, big Hawaiian earrings, and the big free groups

We define for every affine Coxeter graph, a certain factor group of the associated Artin group and prove that some of these groups appear as orbifold fundamental groups of moduli spaces. Examples are the moduli space of nonsingular cubic algebraic surfaces and the universal nonhyperelliptic smooth genus three curve. We use this to obtain a simple presentation of the mapping class group of a compact genus three topological surface with connected boundary. This leads to a modification of Wajnryb's presentation of the mapping class groups in the higher genus case that can be understood in algebro-geometric terms.

We prove that all groups can be realized as fundamental groups of compact spaces if and only if no measurable cardinals exist. If the cardinality of a group $G$ is nonmeasurable then the compact space $K$ such that $G=\pi _1K$ may be chosen so that it is

Observations on the construction of covers using permutation voltage assignments