Graphical Calculus for the Double Affine Q-Dependent Braid Group

In this paper, we present a straightforward pictorial representation of the double affine Hecke algebra (DAHA) which enables us to translate the abstract algebraic structure of a DAHA into an intuitive graphical calculus suitable for a physics audience. Initially, we define the larger double affine Q-dependent braid group. This group is constructed by appending to the braid group a set of operators Q _i , before extending it to an affine Q-dependent braid group. We show specifically that the elliptic braid group and the DAHA can be obtained as quotient groups. Complementing this, we present a pictorial representation of the double affine Q-dependent braid group based on ribbons living in a toroid. We show that in this pictorial representation, we can fully describe any DAHA. Specifically, we graphically describe the parameter q upon which this algebra is dependent and show that in this particular representation q corresponds to a twist in the ribbon.     

Ropelengths of closed braids

For a link K, let L(K) denote the ropelength of K and let Cr(K) denote the crossing number of K. An important problem in geometric knot theory concerns the bound on L(K) in terms of Cr(K). It is well known that there exist positive constants c₁, c₂ such that for any link K, c₁⋅(Cr(K))^3/4?L(K)?c₂⋅(Cr(K))^3/2. In this paper, we show that any closed braid with n crossings can be realized by a unit thickness rope of length at most of the order O(n6/5). Thus, if a link K admits a closed braid representation in which the number of crossings is bounded by a(Cr(K)) for some constant a?1, then we have L(K)?c⋅(Cr(K))^6/5 for some constant c>0 which only depends on a. In particular, this holds for any link that admits a reduced alternating closed braid representation, or any link K that admits a regular projection in which there are at most O(Cr(K)) crossings and Seifert circles.     

A special class of tensor categories initiated by inverse braid monoids

In this paper, we introduce the concept of a wide tensor category which is a special class of a tensor category initiated by the inverse braid monoids recently investigated by Easdown and Lavers [The Inverse Braid Monoid, Adv. in Math. 186 (2004) 438-455].The inverse braid monoidsIB_n is an inverse monoid which behaves as the symmetric inverse semigroup so that the braid group B_n can be regarded as the braids acting in the symmetric group. In this paper, the structure of inverse braid monoids is explained by using the language of categories. A partial algebra category, which is a subcategory of the representative category of a bialgebra, is given as an example of wide tensor categories. In addition, some elementary properties of wide tensor categories are given. The main result is to show that for every strongly wide tensor category C, a strict wide tensor category C^str can be constructed and is wide tensor equivalent to C with a wide tensor equivalence F.As a generalization of the universality property of the braid category B, we also illustrate a wide tensor category through the discussion on the universality of the inverse braid category IB (see Theorem 3.3, 3.6 and Proposition 3.7).     

Positive open book decompositions of Stein fillable 3-manifolds along prescribed links

It is known by Loi and Piergallini that a closed, oriented, smooth 3-manifold is Stein fillable if and only if it has a positive open book decomposition. In the present paper we will show that for every link L in a Stein fillable 3-manifold there exists an additional knot L^′ to L such that the link L∪L^′ is the binding of a positive open book decomposition of the Stein fillable 3-manifold. To prove the assertion, we will use the divide, which is a generalization of real morsification theory of complex plane curve singularities, and 2-handle attachings along Legendrian curves.     

The braid group of a necklace

We show several geometric and algebraic aspects of a necklace: a link composed with a core circle and a series of (unlinked) circles linked to this core. We first prove that the fundamental group of the configuration space of necklaces (that we will call braid group of a necklace) is isomorphic to the braid group over an annulus quotiented by the square of the center. We then define braid groups of necklaces and affine braid groups of type \(\mathcal {A}\) in terms of automorphisms of free groups and characterize these automorphisms among all automorphisms of free groups. In the case of affine braid groups of type \(\mathcal {A}\) such a representation is faithful.     

ON QUOTIENTS OF BRAID GROUPS

ABSTRACT.

Let G be the group ?[t, t ^?1] x ?. By studying the action of the braid group B_n on the set Gⁿ , we obtain representations of B_n into a wreath product of the symmetric group and the general linear group over ?[t, t ^?1]. This in particular recovers the Burau representation of the braid group. Furthermore, some quotients of the braid group are obtained by using the representations found.     

Properly discontinuous actions of subgroups in amenable algebraic groups and its application to affine motions

This note will concern properly discontinuous actions of subgroups in real algebraic groups on contractible manifolds. Let (π,X,ρ) be such an action, where ρ:π→ Diff(X) is a homomorphism. We assume that ? extends to a smooth action of a real algebraic group G containing π. If such π has a nontrivial radical (i.e., unique maximal normal solvable subgroup), then we can apply the method of Seifert construction [14],[17] to yield that the quotient π\X supports the structure of an injective Seifert fibering with typical (resp. exceptional) fiber diffeomorphic to a solv (resp. infrasolv)-manifold (when π acts freely). When G is an amenable algebraic group, we can say about a uniqueness property for such actions. Namely, let (π_i, X_i, ρ_i) be actions as above (i= 1,2). Then, given an isomorphism f of π₁ onto ?₂, there is a diffeomorphism h: X₁→X₂ such that h(ρ₁(r)x)=ρ₂(f(r)h(x).As an application, we try to decide the structure of affine motions of some euclidean space $\text{R}$ⁿ. First we verify the conjecture of [17, 4 5], i.e., a compact complete affinely flat manifold admits a maximal toral action if its fundamental group has a nontrivial center. Second, a compact complete affinity flat manifold whose fundamental group is virtually polycyclic supports the structure of an infrasolvmanifold. This structure varies depending on its solvable kernel (if it is abelian or nilpotent, it must be a euclidean space form or an infranilmanifold respectively). If a group of the affine group A(n) acts properly discontinuously and with compact quotient of $\text{R}$ⁿ, then it is called an affine crystallographic group. Finally, we can say so far as to a uniqueness property that two virtually polycyclic affine crystallographic groups are conjugate inside Diff($\text{R}$ⁿ) if they are isomorphic (cf.[8]).     

Local representations of braid groups

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.     

On endomorphism monoids in varieties of bands

It will be proved that every non-trivial variety \({\mathbb{V}}\) of bands (idempotent semigroups) contains a proper generating class of non-isomorphic bands B such that B generates \({\mathbb{V}}\) and any band \({B\prime}\) having the same endomorphism monoid as B is isomorphic to B or to the opposite band B^op. Consequently, every sharply greater band variety has a sharply greater class of endomorphism monoids.     

Surface framed braids

In this paper we introduce the framed pure braid group on n strands of an oriented surface, a topological generalisation of the pure braid group P _n . We give different equivalent definitions for framed pure braid groups and we study exact sequences relating these groups with other generalisations of P _n , usually called surface pure braid groups. The notion of surface framed braid groups is also introduced.     

L'anneau de cohomologie d'une variété de Seifert

All manifolds M considered in this Note are orientable Seifert 3-manifolds with base surface S² and infinite fundamental group π₁ (M). Our goal is to compute the cohomology ring H^* (M; Z/2Z). The ring structure will enable us to determine whether M admits a degree 1 map into RP³ or not. We describe the equivariant chain complex for the universal cover M of M, and give a diagonal approximation. The cohomology ring H^* (M; Z/2Z) is computed.     

Seifert fibered surgeries with distinct primitive/Seifert positions

We call a pair (K,m) of a knot K in the 3-sphere S³ and an integer m a Seifert fibered surgery if m-surgery on K yields a Seifert fiber space. For most known Seifert fibered surgeries (K,m), K can be embedded in a genus 2 Heegaard surface of S³ in a primitive/Seifert position, the concept introduced by Dean as a natural extension of primitive/primitive position defined by Berge. Recently Guntel has given an infinite family of Seifert fibered surgeries each of which has distinct primitive/Seifert positions. In this paper we give yet other infinite families of Seifert fibered surgeries with distinct primitive/Seifert positions from a different point of view.     

Harmonic analysis of SL(2) over a locally compact field

We carry over the pioneer work of Kunze and Stein concerning representation theory and harmonic analysis on SL(2, R) to the group G = SL(2, K), K a locally compact totally disconnected nondiscrete field. The main result is that convolution by an L^p(G) function, 1 ? p < 2, is a bounded operator on L²(G). To accomplish this result we develop the appropriate estimates (which depend upon the work of Sally et al.) that enable us to apply the Kunze and Stein interpolation theory to the Fourier-Laplace transform for the group G. Best possible estimates are obtained.     

Higher-order polynomial invariants of 3-manifolds giving lower bounds for the Thurston norm

We define an infinite sequence of new invariants, δ_n, of a group G that measure the size of the successive quotients of the derived series of G. In the case that G is the fundamental group of a 3-manifold, we obtain new 3-manifold invariants. These invariants are closely related to the topology of the 3-manifold. They give lower bounds for the Thurston norm which provide better estimates than the bound established by McMullen using the Alexander norm. We also show that the δ_n give obstructions to a 3-manifold fibering over S¹ and to a 3-manifold being Seifert fibered. Moreover, we show that the δ_n give computable algebraic obstructions to a 4-manifold of the form X×S¹ admitting a symplectic structure even when the obstructions given by the Seiberg-Witten invariants fail. There are also applications to the minimal ropelength and genera of knots and links in S³.     

On hyperbolicity of SU(2)-equivariant, punctured disc bundles over the complex affine quadric

Given a holomorphic line bundle over the complex affine quadric Q ², we investigate its Stein, SU(2)-equivariant disc bundles. Up to equivariant biholomorphism, these are all contained in a maximal one, say ??_max. By removing the zero section from ??_max one obtains the unique Stein, SU(2)-equivariant, punctured disc bundle over Q ² which contains entire curves. All other such punctured disc bundles are shown to be Kobayashi hyperbolic.     

Vines and partial transformations

We introduce the partial vine monoid PVn. This monoid is related to the partial transformation semigroup PTn in the same way as the braid group Bn is related to the symmetric group Sn, and contains both the vine monoid [T.G. Lavers, The theory of vines, Comm. Algebra 25 (4) (1997) 1257-1284] and the inverse braid monoid [D. Easdown, T.G. Lavers, The inverse braid monoid, Adv. Math. 186 (2) (2004) 438-455]. We give a presentation for PVn in terms of generators and relations, as well as a faithful representation in a monoid of endomorphisms of a free group. We also derive a new presentation for PTn.     

A Strong Oka Principle for Embeddings of Some Planar Domains into ℂ×ℂ∗

Gromov, in his seminal 1989 paper on the Oka principle, introduced the notion of an elliptic manifold and proved that every continuous map from a Stein manifold to an elliptic manifold is homotopic to a holomorphic map. We show that a much stronger Oka principle holds in the special case of maps from certain open Riemann surfaces called circular domains into ?×?^?, namely that every continuous map is homotopic to a proper holomorphic embedding. An important ingredient is a generalization to ?×?^? of recent results of Wold and Forstneri? on the long-standing problem of properly embedding open Riemann surfaces into ?², with an additional result on the homotopy class of the embeddings. We also give a complete solution to a question that arises naturally in Lárusson’s holomorphic homotopy theory, of the existence of acyclic embeddings of Riemann surfaces with abelian fundamental group into 2-dimensional elliptic Stein manifolds.     

A duality principle for groups

The duality principle for Gabor frames states that a Gabor sequence obtained by a time-frequency lattice is a frame for L²(Rd) if and only if the associated adjoint Gabor sequence is a Riesz sequence. We prove that this duality principle extends to any dual pairs of projective unitary representations of countable groups. We examine the existence problem of dual pairs and establish some connection with classification problems for II₁ factors. While in general such a pair may not exist for some groups, we show that such a dual pair always exists for every subrepresentation of the left regular unitary representation when G is an abelian infinite countable group or an amenable ICC group. For free groups with finitely many generators, the existence problem of such a dual pair is equivalent to the well-known problem about the classification of free group von Neumann algebras.