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This study investigated the effects of different braid materials (polyethylene terephthalate, nylon 6, and their hybrid), weaving frequencies of the braids (7, 9, 11, 13 Hz), and concentrations of polyether sulfone solution (15% and 18 wt%) on braid angles, braid thickness, and bursting resistance. The results showed that for all types of single and hybrid braids, the braid angle decreases with increasing spinning frequency. When spinning frequency was increased from 7 to 13 Hz, the average wall thickness of polyethylene terephthalate, nylon 6, and the hybrid braids increased by approximately 50%, 65%, and 60%, respectively. The bursting pressure results were obtained in the range of 9.3 to 20.2 bars. The Fourier transform infrared spectroscopy results revealed no chemical bond between the braids and the polyether sulfone layer. Optical and electron microscope analyses revealed formation of uniform thickness of the polyether sulfone layers on the surface of the braids, as well as the penetration of polyether sulfone into the gaps between the filaments. The analyses also showed macroscopic adhesion of polyether sulfone to the braids and provided information on the amount and size of the pores in the polymer layers. The results showed that the selection of appropriate weaving frequency, braid materials, and polyether sulfone layer concentration are important as controllable parameters for producing polymeric hollow fiber composites with excellent burst pressure resistance. [ABSTRACT FROM AUTHOR]

Let n, m ∈ ℕ, n ≥ 3 and n ≥ m ≥ 2. In this paper, we study some properties of the unrestricted virtual singular braid of braid monoid UVSBmn and group UVSGmn. These properties emerged by studying the analogous properties of the unrestricted virtual singular pseudosymmetric braid of braid monoid and group. These mathematical objects are submonoids and subgroups of the virtual singular braid group VSGn. For many of these quotients, we have obtained reduced presentations.

A classical theorem of braid theory, dating back to Artin's work, says that two closed braids in a solid torus are ambient isotopic if and only if they represent the same conjugacy class of the braid group. This theorem can be reformulated in the framework of link theory without referring to the group structure. A link in a surface bundle over the circle is transversal whenever it covers the circle. In this terminology, Artin's theorem states that in a solid torus trivially fibered over the circle transversal links are ambient isotopic if and only if they are isotopic in the class of transversal links. We generalize this result by proving that (in the piecewise linear category) transversal links in an arbitrary compact orientable -manifold fibered over the circle with a compact fiber are ambient isotopic if and only if they are isotopic in the class of transversal links. [ABSTRACT FROM AUTHOR]

In the last several decades, textile-reinforced composites have emerged as a unique class of materials offering intricate features, reduced fabrication costs, introduced multiaxial reinforcement, and enhanced damaged tolerance. Despite these benefits, textile-reinforced composites face challenges as predicting their performance often relies on heuristics and past experiences without gaining insights into the underlying structure of the textile material and its constituents. This tutorial-based mini-review aims to delve into the fundamentals of textile architecture in the context of textile-reinforced composites and provide an overview of their significant physical and structural features that influence the performance characteristics of textile-reinforced composites. [ABSTRACT FROM AUTHOR]

Using the recoupling theory, we define a representation of the pure braid group and show that it is not trivial. [ABSTRACT FROM AUTHOR]

The complex problem of ropes (cyclic) bending over sheaves (CBOS) is considered in this article. Ropes that are used to manoeuvre and regulate sails undergo a variety of dynamic stresses which are crucial yet difficult to determine, due to the complexity of the multi-scaled braiding structure of ropes. A full dynamic Finite Element Method (FEM) simulation of a braided rope bending over a sheave is conducted with the commercial software Abaqus/Explicit, as it can capture the complex processes occurring inside the rope to high accuracy. A theoretical deduction is performed to determine the yarn paths in a 3D braided structure. Relative movement between braided yarns (including the interlacing point) are better assessed at yarn level through 3D element types instead of the traditional beam elements, as they allow to obtain an estimation of yarn-yarn displacement. Parametric studies are conducted to better understand the factors that affect the yarn sliding. The ratio between the diameter of the sheave and the rope, and the braiding angle, are found to be crucial for the relative movement, while the number of yarns and their diameters have little influence. The successful simulation process demonstrates the feasibility of modelling and analysing the complex interaction occurring in braided structures under specific boundary conditions. The calculated estimated displacement can contribute to further investigations such as the friction and heat generation problem in braided ropes. Simulation with this method can provide alternatives in predicting lifespan of braided products that cannot be easily inspected. [ABSTRACT FROM AUTHOR]

Representations of braid group B n on n ≥ 2 strands by automorphisms of a free group of rank n go back to Artin. In 1991, Kauffman introduced a theory of virtual braids, virtual knots, and links. The virtual braid group V B n on n ≥ 2 strands is an extension of the classical braid group B n by the symmetric group S n . In this paper, we consider flat virtual braid groups F V B n on n ≥ 2 strands and construct a family of representations of F V B n by automorphisms of free groups of rank 2 n . It has been established that these representations do not preserve the forbidden relations between classical and virtual generators. We investigated some algebraic properties of the constructed representations. In particular, we established conditions of faithfulness in case n = 2 and proved that the kernel contains a free group of rank two for n ≥ 3 . [ABSTRACT FROM AUTHOR]

The complex problem of ropes (cyclic) bending over sheaves (CBOS) is considered in this article. Ropes that are used to manoeuvre and regulate sails undergo a variety of dynamic stresses which are crucial yet difficult to determine, due to the complexity of the multiscaled braiding structure of ropes. A full dynamic Finite Element Method (FEM) simulation of a braided rope bending over a sheave is conducted with the commercial software Abaqus/Explicit, as it can capture the complex processes occurring inside the rope to high accuracy. A theoretical deduction is performed to determine the yarn paths in a 3D braided structure. Relative movement between braided yarns (including the interlacing point) are better assessed at yarn level through 3D element types instead of the traditional beam elements, as they allow to obtain an estimation of yarn-yarn displacement. Parametric studies are conducted to better understand the factors that affect the yarn sliding. The ratio between the diameter of the sheave and the rope, and the braiding angle, are found to be crucial for the relative movement, while the number of yarns and their diameters have little influence. The successful simulation process demonstrates the feasibility of modelling and analysing the complex interaction occurring in braided structures under specific boundary conditions. The calculated estimated displacement can contribute to further investigations such as the friction and heat generation problem in braided ropes. Simulation with this method can provide alternatives in predicting lifespan of braided products that cannot be easily inspected. [ABSTRACT FROM AUTHOR]

We prove the first polynomial bound on the number of monotonic homotopy moves required to tighten a collection of closed curves on any compact orientable surface, where the number of crossings in the curve is not allowed to increase at any time during the process. The best known upper bound before was exponential, which can be obtained by combining the algorithm of De Graaf and Schrijver [J. Comb. Theory Ser. B, 1997] together with an exponential upper bound on the number of possible surface maps. To obtain the new upper bound, we apply tools from hyperbolic geometry, as well as operations in graph drawing algorithms—the cluster and pipe expansions—to the study of curves on surfaces. As corollaries, we present two efficient algorithms for curves and graphs on surfaces. First, we provide a polynomial-time algorithm to convert any given multicurve on a surface into minimal position. Such an algorithm only existed for single closed curves, and it is known that previous techniques do not generalize to the multicurve case. Second, we provide a polynomial-time algorithm to reduce any k-terminal plane graph (and more generally, surface graph) using degree-1 reductions, series-parallel reductions, and Δ Y-transformations for arbitrary integer k. Previous algorithms only existed in the planar setting when k ≤ 4, and all of them rely on extensive case-by-case analysis based on different values of k. Our algorithm makes use of the connection between electrical transformations and homotopy moves and thus solves the problem in a unified fashion. [ABSTRACT FROM AUTHOR]

Representations of braid group Bn on n≥2 strands by automorphisms of a free group of rank n go back to Artin. In 1991, Kauffman introduced a theory of virtual braids, virtual knots, and links. The virtual braid group VBn on n≥2 strands is an extension of the classical braid group Bn by the symmetric group Sn. In this paper, we consider flat virtual braid groups FVBn on n≥2 strands and construct a family of representations of FVBn by automorphisms of free groups of rank 2n. It has been established that these representations do not preserve the forbidden relations between classical and virtual generators. We investigated some algebraic properties of the constructed representations. In particular, we established conditions of faithfulness in case n=2 and proved that the kernel contains a free group of rank two for n≥3.

To modulate the physicochemical features of poly(ethylene terephthalate) (PET) multifilaments surface composing a complex textile structure (core and shell system), intended to improve upon current implants for high extension injuries of the Achilles tendon or even for its total replacement, two surface treatments with different purposes (bioactive and biopassive) were studied. The first treatment is based on amino groups grafting using ethylenediamine molecules to be applied in the structure core to improve cell adhesion and proliferation. The other treatment relates to a polytetrafluoroethylene (PTFE) coating to be applied in the structure shell to decrease its coefficient of friction and avoid adhesions. Both treatments were optimized to reach their purposed goals without harming the tensile properties of PET yarns, which were evaluated by static tensile tests. The resazurin assay and scanning electron microscopy analysis showed that the purposed goals related to fibroblast adhesion were achieved for both treatments and in the case of PTFE coating, the abrasion resistance was also improved according to the yarn‐on‐yarn abrasion tests. [ABSTRACT FROM AUTHOR]

In this paper, we show that every finite spatial graph is a connected sum of a planar graph, which is a forest, and a tangle. As a consequence, we get that any finite spatial graph is a connected sum of a planar graph and a braid. Using these decompositions it is not difficult to find a set of generators and defining relations for the fundamental group of compliment of a spatial graph in 3-space ℝ 3 . [ABSTRACT FROM AUTHOR]

Recently, the first named author defined a 2-parametric family of groups Gnk [V. O. Manturov, Non-reidemeister knot theory and its applications in dynamical systems, geometry and topology, preprint (2015), arXiv:1501.05208]. Those groups may be regarded as analogues of braid groups. Study of the connection between the groups Gnk and dynamical systems led to the discovery of the following fundamental principle: "If dynamical systems describing the motion of n particles possess a nice codimension one property governed by exactly kk particles, then these dynamical systems admit a topological invariant valued in Gnk". The Gnk groups have connections to different algebraic structures, Coxeter groups, Kirillov-Fomin algebras, and cluster algebras, to name three. Study of the Gnk groups led to, in particular, the construction of invariants, valued in free products of cyclic groups. All generators of the Gnk groups are reflections which make them similar to Coxeter groups and not to braid groups. Nevertheless, there are many ways to enhance Gnk groups to get rid of this 22-torsion. Later the first and the fourth named authors introduced and studied the second family of groups, denoted by Γnk, which are closely related to triangulations of manifolds. The spaces of triangulations of a given manifolds have been widely studied. The celebrated theorem of Pachner [P.L. homeomorphic manifolds are equivalent by elementary shellings, Europ. J. Combin.12(2) (1991) 129-145] says that any two triangulations of a given manifold can be connected by a sequence of bistellar moves or Pachner moves. See also [I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants (Birkhäuser, Boston, 1994); A. Nabutovsky, Fundamental group and contractible closed geodesics, Comm. Pure Appl. Math.49(12) (1996) 1257-1270]; the Γnk naturally appear when considering the set of triangulations with the fixed number of points. There are two ways of introducing the groups Γnk: the geometrical one, which depends on the metric, and the topological one. The second one can be thought of as a "braid group" of the manifold and, by definition, is an invariant of the topological type of manifold; in a similar way, one can construct the smooth version. In this paper, we give a survey of the ideas lying in the foundation of the Gnk and Γnk theories and give an overview of recent results in the study of those groups, manifolds, dynamical systems, knot and braid theories. [ABSTRACT FROM AUTHOR]

Recently, the first named author defined a 2-parametric family of groups Gnk [V. O. Manturov, Non–reidemeister knot theory and its applications in dynamical systems, geometry and topology, preprint (2015), arXiv:1501.05208]. Those groups may be regarded as analogues of braid groups. Study of the connection between the groups Gnk and dynamical systems led to the discovery of the following fundamental principle: “If dynamical systems describing the motion of n particles possess a nice codimension one property governed by exactly k particles, then these dynamical systems admit a topological invariant valued in Gnk”. The Gnk groups have connections to different algebraic structures, Coxeter groups, Kirillov-Fomin algebras, and cluster algebras, to name three. Study of the Gnk groups led to, in particular, the construction of invariants, valued in free products of cyclic groups. All generators of the Gnk groups are reflections which make them similar to Coxeter groups and not to braid groups. Nevertheless, there are many ways to enhance Gnk groups to get rid of this 2-torsion. Later the first and the fourth named authors introduced and studied the second family of groups, denoted by Γnk, which are closely related to triangulations of manifolds. The spaces of triangulations of a given manifolds have been widely studied. The celebrated theorem of Pachner [P.L. homeomorphic manifolds are equivalent by elementary shellings, Europ. J. Combin. 12(2) (1991) 129–145] says that any two triangulations of a given manifold can be connected by a sequence of bistellar moves or Pachner moves. See also [I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants (Birkhäuser, Boston, 1994); A. Nabutovsky, Fundamental group and contractible closed geodesics, Comm. Pure Appl. Math. 49(12) (1996) 1257–1270]; the Γnk naturally appear when considering the set of triangulations with the fixed number of points. There are two ways of introducing the groups Γnk: the geometrical one, which depends on the metric, and the topological one. The second one can be thought of as a “braid group” of the manifold and, by definition, is an invariant of the topological type of manifold; in a similar way, one can construct the smooth version. In this paper, we give a survey of the ideas lying in the foundation of the Gnk and Γnk theories and give an overview of recent results in the study of those groups, manifolds, dynamical systems, knot and braid theories.

• This paper reviews various aspects of reinforced hollow fiber membranes (HFMs). • There is a focus on the braid-reinforced hollow fiber membranes. • Advantages and disadvantages of reinforcement methods are presented. • Tensile strength and elongation at breaks are the most studied properties of HFMs. • Some crucial issues of reinforced hollow fiber membranes need to be further studied. Reinforcing the HFMs provides appropriate mechanical and shear strengths and improves break at elongation. This comprehensive review paper first discusses the fundamental concept of HFMs fabrications, advantages and disadvantages of HFMs, fabrication methods and fabrication materials. Then, reinforcement methods are categorized into porous matrix membrane reinforced method, the fibers (use of threads or filament) reinforced method, and the use of tubular braids, and each one is discussed based on the concept of thermochemical compatibility of the supporting material and the base polymer. Such a compatibility leads to homogeneous- and heterogeneous-reinforcement method with different interfacial bonding state. A comparative study on the mentioned reinforcement methods is also presented based on the tabulated and charted tensile strength and elongation at break data. At the end, a brief look is taken into commercialized products and worldwide projects installed or under installation in the field of reinforced HFMs. [Display omitted] [ABSTRACT FROM AUTHOR]

A common braiding machine cannot perform continuous braiding using closed annular axis mandrels. To solve this problem, a modified vertical braiding machine was made to braid composite preforms with irregular cross-section mandrels. The finite element method was used to simulate the braiding process, and an efficient method was also derived to predict the braiding angles. The results show that the predicted braiding angles are basically consistent with the actual braiding angles, and the braiding angles at distinctive locations on the braided preform recorded differences of up to 10° or more than 30%. Braiding process simulation via the finite element method can thus effectively and vividly reflect the yarn path on the preform. As such, the braiding angles on the braided preforms can be realized through projection and surface flattening with much better accuracy. It also resolves the difficult problem often faced in measuring the braiding angles at the corner of the mandrel and provides a solid basis for continued research on the performance of its composite reinforcement. [ABSTRACT FROM AUTHOR]

A transnational feminist paradigm guides this inquiry when braiding narratives, both literally and abstractly, as a feminist methodology for data analysis. The data include personal narratives (memories and thoughts imprinted on my body, spirit, and mind), master narratives (collaborative databases and public sites), and counter narratives (articles, books, and videos). The braid offers opportunities to (re)imagine coalitional possibilities through figures (and) that materialize in the tension of weaving together. When foregrounding self-reflection in the context of weaving together, I explore supremacy, positionality, and representation. The assemblage of figures leads me to a colleague turned friend turned collaborator who reminds me to work with productive tension toward love. [ABSTRACT FROM AUTHOR]

In the present paper we construct braid group actions on quantum symmetric pair coideal subalgebras of type AIII/AIV. This completes the proof of a conjecture by Kolb and Pellegrini in the case where the underlying Lie algebra is sl n. The braid group actions are defined on the generators of the coideal subalgebras and the defining relations and braid relations are verified by explicit calculations. [ABSTRACT FROM AUTHOR]

In this paper, we study representations of G n 3 -like groups. The group G n 3 itself appeared in works of the third named author on non-Reidemeister knot (and braid) theory. This group is closely related to dynamical systems of points and their invariants. Representations of G n 3 -like groups are useful both for the study of those groups themselves, and constructing invariants of knots and braids based on the G n 3 -like group structure. [ABSTRACT FROM AUTHOR]

Herein, a theoretical analysis of auxetic tubular braid (ATB) formed with three types of component yarns having different diameter and modulus is presented. A simple theoretical model is developed for predicting the deformation behavior of this structure under uniaxial extension. With its accuracy first validated by the experimental data, the developed model is further corrected based on the real condition and thus becomes capable to characterize the auxetic behavior of the ATB. During the analysis, it is found that the stiff wrap yarn cannot be fully straightened in real condition and its helical angle decreasing rate is inconsistent during the stretching. Herein, an effective tool for understanding the deformation behavior is provided, which is meaningful for the braid's improved design and applications. [ABSTRACT FROM AUTHOR]

Historic Japanese textiles from over 1000 years ago generally show marked deterioration and only very rare examples show their original forms and much information about textile reproduction has been lost. The replication of textile braids lacks systematic methodology and is still being practiced by only few individual braiding experts. The recreation of original braids as close as possible to original braids is a part of Japan's intangible cultural heritage. The aim of this study is to clarify the decision-making procedure through which the braiding experts can decipher the original braiding structures. As a preliminary study of this project, interviews of a braid researcher and a replicating expert, Makiko Tada were performed regarding her working practices. It is important to clarify the braiding parameters for structural analysis such as the number of transits and the balance of ridges, and it became clear that the orientation of multiple colored threads plays an important role. The expert's replicate works were also analyzed using a text-mining statistical technique to clarify the relationship of braiding parameters. The relationship between each braiding parameter and production method such as loop manipulation and Taka-dai became clear. As a result, the process of deciphering the original braid structure has been compiled in simplified workflows, which could contribute to the standardization and improvement in efficiency of replication of cultural property braids. [ABSTRACT FROM AUTHOR]

This article deals with braid groups of complex reflection groups as introduced in [2]. We present results in two directions. First, we extend the works of Gonçalves, Guaschi, Ocampo [5] and Marin [7] on finite subgroups of the quotients of generalized braid groups by the derived subgroup of their pure braid group. We get explicit criteria for subgroups of the (complex) reflection group to lift to subgroups of this quotient. In the specific case of the classical braid group, this enables us to describe all its finite subgroups: we show that every odd-order finite group can be embedded in it, when the number of strands goes to infinity. We also determine a complete list of the irreducible reflection groups for which this quotient is a Bieberbach group. In the second part, we describe the abelianization of the inverse image of a subgroup of a reflection group in its generalized braid group generalizing results of the first author [1]. In particular, these abelianized groups are not necessarily torsion-free. [ABSTRACT FROM AUTHOR]

Let Γg,1m be the mapping class group of the orientable surface Σg,1m of genus g with one parametrized boundary curve and m permutable punctures; when m = 0 we omit it from the notation. Let βm(Σg,1) be the braid group on m strands of the surface Σg,1. We prove that H*(Γg,1m; Z2) ≅ H*(Γg,1; H*(βm(Σg,1); Z2)). The main ingredient is the computation of H*(βm(Σg,1); Z2) as a symplectic representation of Γg,1. [ABSTRACT FROM AUTHOR]

We construct taut foliations in every closed 3-manifold obtained by r-framed Dehn surgery along a positive 3-braid knot K in S³, where r < 2g(K) - 1 and g(K) denotes the Seifert genus of K. This confirms a prediction of the L-space Conjecture. For instance, we produce taut foliations in every non-L-space obtained by surgery along the pretzel knot P(-2, 3, 7), and indeed along every pretzel knot P(-2, 3, q), for q a positive odd integer. This is the first construction of taut foliations for every non-L-space obtained by surgery along an infinite family of hyperbolic L-space knots. Additionally, we construct taut foliations in every closed 3-manifold obtained by r -framed Dehn surgery along a positive 1-bridge braid in S³, where r < g(K). This paper relies extensively on color figures. Some references to colour may not be meaningful in the printed version, and we refer the reader to the online version which includes the color figures. [ABSTRACT FROM AUTHOR]

An inversion triple of an element w of a simply laced Coxeter group W is a set { α , β , α + β } , where each element is a positive root sent negative by w. We say that an inversion triple of w is contractible if there is a root sequence for w in which the roots of the triple appear consecutively. Such triples arise in the study of the commutation classes of reduced expressions of elements of W, and have been used to define or characterize certain classes of elements of W, e.g., the fully commutative elements and the freely braided elements. Also, the study of inversion triples is connected with the representation theory of affine Hecke algebras and double affine Hecke algebras. In this paper, we describe the inversion triples that are contractible, and we give a new, simple characterization of the groups W with the property that all inversion triples are contractible. We also study the natural action of W on the set of all triples of (not necessarily positive) roots of the form { α , β , α + β } . This enables us to prove rather quickly that every triple of positive roots { α , β , α + β } is contractible for some w in W and, moreover, when W is finite, w may be taken to be the longest element of W. At the end of the paper, we pose a problem concerning the aforementioned action. [ABSTRACT FROM AUTHOR]

Bobbin lace is a fibre art form in which threads are braided together to form a fabric, often with a very detailed and complex design. In traditional practice, each region of the fabric is filled with a periodic texture. We establish the groundwork for non-periodic lace patterns and present three new quasiperiodic families based on Sturmian words, the Penrose tiling by thick and thin rhombs and the Ammann-bar decoration of the Penrose tiling. [ABSTRACT FROM AUTHOR]

We prove that certain sequences of Artin monoids containing the braid monoid as a submonoid satisfy homological stability. When the K(π, 1) conjecture holds for the associated family of Artin groups, this establishes homological stability for these groups. In particular, this recovers and extends Arnol'd's proof of stability for the Artin groups of type A, B and D. [ABSTRACT FROM AUTHOR]

In this paper, the main idea is to present the solvability of the word problem of pure virtual braid groups by using Gröbner-Shirshov basis theory. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of this paper is to introduce an innovative transversal tubular braid texture and to study the elastic behavior of its 3 D printed structure comparatively to 3 D printed longitudinal tubular braid texture (maypole) to be used as reinforcement. Design/methodology/approach: Regarding the lack of proper machines for the production of the proposed texture, the structure of samples was produced as a tubular lattice braid texture using a 3 D printer with the fused deposition modeling method subsequent to simulation by Rhinoceros software. The produced specimens were composited by polyurethane resin. The composite samples were evaluated by the split disk mechanical test to obtain their hoop stress. The structures of the reinforced composites were theoretically analyzed by ANSYS software. Findings: The results of the mechanical test and theoretical analysis showed that the composites reinforced with transversal tubular lattice braid have higher strength compared to the composites reinforced with longitudinal ones. This assured that the composite reinforced by transversal tubular lattice braid is reliable to be used as high-performance tube for different applications. Originality/value: Further work is carried out to produce the innovated complex structure continuously by a specially designed machine and fibrous materials to reinforce tubular composites in an industrial continual process to be applied for high-pressure fluids flows. [ABSTRACT FROM AUTHOR]

This paper discusses a man's obsession with the memories of his dead wife as represented in Georges Rodenbach's novel Bruges-la-Morte. More precisely, Rodenbach draws a haunting picture of the woman's hair, which is personified and acquires the dimension of a full-fledged character in the novel. The Belgian author places Hugues Viane dead wife's hair at the center of his particular vision of fictional space, vacillating between tangible and intangible (Boraczek 1999) as well as between sacred and demoniac. The woman's gold braid is preserved in a glass case: by worshipping it, the widower makes a religion of his sorrow. Furthermore, the woman's hair generates a dense web of analogies between Bruges, in which Bachelard (1942) sees the Ophelization of an entire city, Viane's dead wife compared in the novel to a Virgin, and the actress Jane Scoti, a femme fatale who seems to have the same blond hair as the dead woman. By doing so, Rodenbach places the woman's figure at the crossroads between literature and visual arts. [ABSTRACT FROM AUTHOR]