Standard Braid Research Articles

Mapping class groups of covers with boundary and braid group embeddings

We consider finite-sheeted, regular, possibly branched covering spaces of compact surfaces with boundary and the associated liftable and symmetric mapping class groups. In particular, we classify when either of these subgroups coincides with the entire mapping class group of the surface. As a consequence, we construct infinite families of non-geometric embeddings of the braid group into mapping class groups in the sense of Wajnryb. Indeed, our embeddings map standard braid generators to products of Dehn twists about curves forming chains of arbitrary length. As key tools, we use the Birman-Hilden theorem and the action of the mapping class group on a particular fundamental groupoid of the surface.

On a surface singular braid monoid

We introduce a monoid corresponding to knotted surfaces in four space, from its hyperbolic splitting represented by marked diagram in braid like form. It has four types of generators: two standard braid generators and two of singular type. Then we state relations on words that follow from topological Yoshikawa moves. As a direct application we will reprove some known theorem about twist-spun knots. We wish then to investigate an index associated to the closure of surface singular braid. Using our relations we will prove that there are exactly six types of knotted surfaces with the index less or equal to two, and there are infinitely many types of surface-knots with index equal to three. Towards the end we will construct a family of classical diagrams such that to unlink them requires at least four Reidemeister III moves.

POSITIVE PERMUTATION BRAIDS AND PERMUTATION INVERSIONS WITH SOME APPLICATIONS

In this paper we constructed an isomorphic group of binary matrices to a finite symmetric group. Our method is based on the inversion of permutations. Using this embedding we find an algorithm for writing down a standard braid word representation for each positive permutation braid. Also an algorithm for writing basis of Hecke algebra Hn+1from such basis of Hnis given.

Current flow in coaxial braided cable shields

In the last few years, much effort has been made to describe the behavior of shielded cables. Many researchers have attempted to understand how an electromagnetic field couples into a braided coaxial cable. There are some important publications on this topic. Nevertheless, up to now, it has not been possible to predict analytically the coupling through a braid shield. An electromagnetic field outside a cable induces a disturbance current in the cable shield. The coupling from the current in the shield into the cable can be described by the transfer impedance. How the current flows in the cable shield is an important quantity in this coupling process. Therefore, to understand the coupling mechanism into a cable, it is necessary to understand the behavior of the current flow in such a braided shield. The paper discusses the current flow in a braided cable shield. The assumption often made in the literature, that a braided shield behaves like a homogeneous tube with apertures, is shown to be inaccurate. It is also shown that the standard braid of the shield used had the same properties as a braid made with insulated wires.

Modified braid equations, Baxterizations and noncommutative spaces for the quantum groups GLq(N), SOq(N), and Spq(N)

Modified braid equations satisfied by generalized R̂ matrices (for a given set of group relations obeyed by the elements of T matrices) are constructed for q-deformed quantum groups GLq(N), SOq(N), and Spq(N) with arbitrary values of N. The Baxterization of R̂ matrices, treated as an aspect complementary to the modification of the braid equation, is obtained for all these cases in particularly elegant forms. A new class of braid matrices is discovered for the quantum groups SOq(N) and Spq(N). The R̂ matrices of this class, while being distinct from the restrictions of the universal R̂ matrix to the corresponding vector representations, satisfy the standard braid equation. The modified braid equation and the Baxterization are obtained for this new class of R̂ matrices. Diagonalization of the generalized R̂ matrices is studied. The diagonalizers are obtained explicitly for some lower dimensional cases in a convenient way, giving directly the eigenvalues of the corresponding R̂ matrices. Applications of such diagonalization are then studied in the context of associated covariantly quantized noncommutative spaces.

Modified braid equations for SOq(3) and noncommutative spaces

General solutions of the R̂TT equation with a maximal number of free parameters in the specrtal decomposition of vector SOq(3) R̂ matrices are implemented to construct modified braid equations (MBE). These matrices conserve the given, standard, group relations of the nine elements of T, but are not constrained to satisfy the standard braid equation (BE). Apart from q and a normalization factor our R̂ contains two free parameters, instead of only one such parameter for deformed unitary algebras studied in a previous paper [Math. QA/0009178] where the nonzero right hand side of the MBE had a linear term proportional to (R̂(12)−R̂(23)). In the present case the rhs is, in general, nonlinear. Several particular solutions are given (Sec. II) and the general structure is analyzed (Appendix A). Our formulation of the problem in terms of projectors yields also two new solutions of standard (nonmodified) braid equation (Sec. II) which are further discussed (Appendix B). The noncommutative three-spaces obtained by implementing such generalized R̂ matrices are studied (Sec. III). The role of coboundary R̂ matrices (not satisfying the standard BE) is explored. The MBE and Baxterization are presented as complementary facets of the same basic construction, namely, the general solution of R̂TT equation (Sec. IV). A new solution is presented in this context. As a simple but remarkable particular case a nontrivial solution of BE is obtained (Appendix B) for q=1. This solution has no free parameter and is not obtainable by twisting the identity matrix. In the concluding remarks (Sec. V), among other points, generalization of our results to SOq(N) is discussed.

Three-Dimensional Realizations of Braids

Let us consider a standard braid diagram as a three-dimensional figure viewed from the top; what happens when we look at this figure from the side? Then we can obtain a new braid, and studying the connection between the initial braid and the derived braid so obtained provides both a new simple proof for the existence of the right greedy normal form of positive braids and a geometrical interpretation for the automatic structure of the braid groups. AMS Subject classification: 20F36, 57M25

Complexes of non-positive curvature and automorphisms of the 4-punctured sphere

Introduction. In [1], we showed that any extension of the rank two free group F2 by 2g can be made to act cocompactly and by isometries on a 2-complex of non-positive curvature. The proof used the fact that every such extension can be realized as the fundamental group of the mapping torus of an automorphism of the once-punctured torus. In fact, the 2-complex constructed in [t] is a deformation retraction of the mapping torus of the surface automorphism. In this paper we show that for an automorphism of the lbur-punctured sphere the corresponding mapping torus can be deformation retracted to a 2-complex which can also be given a metric of non-positive curvature. Thus an extension of the rank three free group/73 by 7Z which can be realized as the fundamental group of the mapping torus of a four-punctured sphere automorphism can also be made to act cocompactly and by isometries on such a 2-complex. The action can be used to show that, as groups, such extensions have automatic structures, in the sense of [3]. The automorphism group. In [2], the structure of the automorphism group of the four-punctured sphere is described. It is an amalgamation of two finite groups over a subgroup which has index three in one vertex group and index two in the other. The vertex groups are the groups consisting of the symmetries of the first two graphs in Figure 1 (a). We only allow symmetries which can be extended to automorphisms of the four-punctured sphere. For simplicity we will confine our attention to those automorphisms of the four-punctured sphere which fix one of the punctures and which are orientation-preserving. Thus the group of automorphisms which we consider is a free product of Z 3 with Z2. This group is also well-known to be the quotient of the three strand braid group B3 by its centre and has presentation G = (a, blaba = bab, (ab) 3 = 1), where a and b are the standard braid generators. The relationship between the generators is given by x = ab, y = aba and a = x2y, b = yx 2, where x and y are the order three and order two generators, respectively, for Z3 * ZzThe 2-complexes. We now describe how, given an automorphism expressed as a word in the generators a and b, we can build a 2-complex whose fundamental group wilt be F 3 extended by the corresponding free group automorphism. For each letter c e {a, b, & b}

Comparison of Mechanical Deformation Properties of Metallic Stents with Use of Stress-Strain Analysis

Elastic and plastic deformation properties of the Wallstent, Palmaz stent, and Strecker stent were evaluated quantitatively with an in vitro model simulating forces exerted by an eccentric lesion. A miniaturized compression testing device was constructed. Stress-strain graphs were obtained for each stent, and the elastic moduli and yield points were calculated. There is a 21-fold range in the elastic modulus among the Wallstent, Palmaz stent, and Strecker stents. The Palmaz stent was the only device to exhibit permanent plastic deformation. The 10-mm Palmaz stent will undergo 15% focal eccentric narrowing at 0.75 atm of pressure; the "standard braid" and "less shortening braid" 10-mm Wallstents at 0.55 and 0.25 atm, respectively; and the 10-mm tantalum Strecker stent at 0.08 atm. Overlapping of stents doubles the stiffness of the Wallstent and the Strecker stent and doubles the yield point of the Palmaz stent. The 4-9 mm Palmaz stent is 30% more resistant to deformation than the larger 8-12-mm version when expanded to identical 8-mm diameters. The "standard braid" version of the 10-mm Wallstent provides 2.3-fold additional strength for resistant stenoses compared with the "less shortening braid." Overlapping or nesting of stents may permit full expansion should there be incomplete expansion or recoil of a single stent. The 4-9-mm Palmaz stent is preferable from the standpoint of allowing the use of a smaller (7-F instead of 9-F) introducer sheath and also for providing superior resistance to deformation. A purely elastic stent such as the Wallstent is preferable in locations where permanent plastic deformation may occur, such as the thoracic outlet.

STATISTICS AND SPIN ON TWO-DIMENSIONAL SURFACES

The fundamental groups of the configuration spaces for the O(3) nonlinear σ-model on the compact genus g surfaces [Formula: see text] and on the connected sums [Formula: see text] are known for any soliton number N. So are the braid for N spinless particles on these manifolds. The representations of these groups govern the possible statistics of solitons and particles. We show that when spin and creation/annihilation processes are introduced, the fundamental groups for the particles are the same as the corresponding σ-model groups. These fundamental groups incorporate the spin-statistics connection and are of greater physical relevance than the standard braid groups.

The Automorphism Groups of the Braid Groups

In the first of two papers published in the Annals in 1947 [3] Emil Artin mentioned the problem of determining all automorphisms of the braid groups (of the Euclidean plane), and in the second [4] took a first step towards a solution. The main result of this paper is a complete determination of these automorphism groups: the outer automorphism group is of order two, generated by the automorphism class containing the element which inverts each of the standard braid generators. Visually, a plane projection of the automorphic image of a braid m is obtained from one for f by replacing each over-crossing by an under-crossing. This result establishes a recent conjecture of Pietrowski and Solitar, to whom we are indebted for bringing the problem to our attention. They obtained the first determination of the automorphism group of the fourstring braid group, using techniques for computing automorphism groups of amalgamated free products [13]. (The result for braids on at most three strings is easy and seems part of the folklore.) The argument we employ is algebraic, and is patterned after that