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The aim of this study is to review three-dimensional (3D) braided fabrics and, in particular, to provide a critical review of the development of 3D braided preform structures and techniques. 3D braided preforms are classified based on various parameters depending on the yarn sets, yarn orientation and intertwining, micro-meso unit cells and macro geometry. Biaxial and triaxial two-dimensional (2D) braided fabrics have been widely used as simple- and complex-shaped structural composite parts in various technical areas. However, 2D braided fabric has size and thickness limitations. 3D braided fabrics have multiple layers and no delamination due to intertwine-type out-of-plane interlacement. However, the 3D braided fabrics have low transverse properties and they also have size and thickness limitations. On the other hand, various unit cell base models on 3D braiding were developed to analyze the properties of 3D braided structures. Most of the unit cell base models include micromechanics and numerical techniques. Multiaxis 3D braided fabrics have multiple layers and no delamination. The in-plane properties of multiaxis 3D braided fabrics may be enhanced due to the ±bias yarn layers. However, the multiaxis 3D braiding technique is at an early stage of development and needs to be fully automated.

Braids are generally divided into 2D braids and 3D braids. Two-dimensional braids include flat braids and circular braids. Circular braids represent three-dimensional textiles, as they enclose a volume, but consist of a two-dimensional yarn architecture. Three-dimensional braids are defined by a three-dimensional yarn architecture. Historically, 3D braids were produced on row and column braiding machines with Cartesian or radial machine beds, by bobbin movements around inlay yarns. Three-dimensional rotary braiding machines allow a more flexible braiding process, as the bobbins are moved via individually controlled horn gears and switches. Both braiding machines at the Institut für Textiltechnik (ITA) of RWTH Aachen University, Germany, are based on the principal of 3D rotary machines. The fully digitized 3D braiding machine with an Industry 4.0 standard enables the near-net-shape production of three-dimensionally braided textile preforms for lightweight applications. The preforms can be specifically reinforced in all three spatial directions according to the application. Complex 3D structures can be produced in just one process step due to the high degree of design freedom. The 3D hexagonal braiding technology is used in the field of medical textiles. The special shape of the horn gears and their hexagonal arrangement provides the densest packing of the bobbins on the machine bed. In addition, the lace braiding mechanism allows two bobbins to occupy the position between two horn gears, maximizing the number of bobbins. One of the main applications is the near-net-shape production of tubular structures, such as complex stent structures. Three-dimensional braiding offers many advantages compared to 2D braiding, e.g., production of complex three-dimensional geometries in one process step, connection of braided layers, production of cross-section changes and ramifications, and local reinforcement of technical textiles without additional process steps. In the following review, the latest developments in 3D braiding, the machine development of 3D braiding machines, as well as software and simulation developments are presented. In addition, various applications in the fields of lightweight construction and medical textiles are introduced.

Benefiting from the multi-directional load-bearing capability, the three-dimensional braided composites (3DBC) have found a wide application in primary structures. It is therefore of great importance to fully understand their mechanical behavior and failure modes. In the present paper, the tensile and compressive tests were carried out, according to standardized testing methods, for eight types of 3DBC, which were manufactured by resin transfer molding (RTM). It was found that the mechanical properties of the 3DBCs decreased with an increasing braiding angle. When the braiding angle was 20°, 3D 5-directional braided composite (3D5dBC) exhibited the best mechanical properties, while for the braiding angle of 40°, the mechanical properties of 3D6dBC were the most prominent. Moreover, the tensile strength of the 3DBCs is approximately two times as much as the compressive strength; however, the compressive modulus is always 10% higher than the tensile modulus. The failure modes of the 3DBCs with a braiding angle of 20°greatly depended on the braiding structures. However, they tend to be consistent when the braiding angle increases to 40°.

The chiral Majorana fermion is a massless self-conjugate fermion which can arise as the edge state of certain 2D topological matters. It has been theoretically predicted and experimentally observed in a hybrid device of a quantum anomalous Hall insulator and a conventional superconductor. Its closely related cousin, the Majorana zero mode in the bulk of the corresponding topological matter, is known to be applicable in topological quantum computations. Here we show that the propagation of chiral Majorana fermions leads to the same unitary transformation as that in the braiding of Majorana zero modes and propose a platform to perform quantum computation with chiral Majorana fermions. A Corbino ring junction of the hybrid device can use quantum coherent chiral Majorana fermions to implement the Hadamard gate and the phase gate, and the junction conductance yields a natural readout for the qubit state.

We study taut foliations on the complements of non-split positive braid closures in \\(S^3\\). If \\(L\\) is such a link with components \\(L_1,\\ldots,L_n\\) and at least one component is not the unknot, then the Dehn surgery along a multislope \\((s_1,\\ldots,s_n)\\in\\mathbb{Q}^n\\) satisfying \\(s_i<2g(L_i)-1\\) for \\(i=1,2,\\ldots, n\\) yields a non-L-space that admits a co-oriented taut foliation.

Non-Abelian topological order is a coveted state of matter with remarkable properties, including quasiparticles that can remember the sequence in which they are exchanged 1 – 4 . These anyonic excitations are promising building blocks of fault-tolerant quantum computers 5 , 6 . However, despite extensive efforts, non-Abelian topological order and its excitations have remained elusive, unlike the simpler quasiparticles or defects in Abelian topological order. Here we present the realization of non-Abelian topological order in the wavefunction prepared in a quantum processor and demonstrate control of its anyons. Using an adaptive circuit on Quantinuum’s H2 trapped-ion quantum processor, we create the ground-state wavefunction of D 4 topological order on a kagome lattice of 27 qubits, with fidelity per site exceeding 98.4 per cent. By creating and moving anyons along Borromean rings in spacetime, anyon interferometry detects an intrinsically non-Abelian braiding process. Furthermore, tunnelling non-Abelions around a torus creates all 22 ground states, as well as an excited state with a single anyon—a peculiar feature of non-Abelian topological order. This work illustrates the counterintuitive nature of non-Abelions and enables their study in quantum devices. A trapped-ion quantum processor is used to create ground-states and excitations of non-Abelian topological order on a kagome lattice of 27 qubits with high fidelity.

Throughout high school, two things that occupied a lot of her time were sports, including braiding hair for her teammates, and picking rocks. With on-the-job training and an inquisitive nature as her primary source of education, she now does everything from herd health to computer work and payroll and helps in the field if needed. Once-a-week bedding changes and storing the sand on concrete directly influenced both production and milk quality.

We determine when a Legendrian quasipositive 3-braid closure in standard contact \\(\\mathbb{R}^3\\) admits an orientable or non-orientable exact Lagrangian filling. Our main result provides evidence for the orientable fillability conjecture of Hayden and Sabloff, showing that a 3-braid closure is orientably exact Lagrangian fillable if and only if it is quasipositive and the HOMFLY bound on its maximum Thurston-Bennequin number is sharp. Of possible independent interest, we construct explicit Legendrian representatives of quasipositive 3-braid closures with maximum Thurston-Bennequin number.

We study a specific line arrangement obtained from a generic \\(2\\)-section of the braid arrangement, and compute the fundamental group of its complement via braid monodromy. We show that the resulting presentation of the fundamental group coincides, under the identification of generators, with the modified Artin presentation introduced by Margalit and McCammond. Moreover, we extend the construction to the Manin--Schechtman arrangements \\(MS(n, k)\\), which are higher analogues of the braid arrangement. Focusing on the case \\(k = 2\\), we obtain an explicit presentation of \\(\\pi_1(\\mathbb{C}^n \\setminus MS(n, 2))\\).

Anyons are quasiparticles that, unlike fermions and bosons, show fractional statistics when two of them are exchanged. Here, we report the experimental observation of anyonic braiding statistics for the ν = 1/3 fractional quantum Hall state by using an electronic Fabry–Perot interferometer. Strong Aharonov–Bohm interference of the edge mode is punctuated by discrete phase slips that indicate an anyonic phase θ anyon = 2π/3. Our results are consistent with a recent theory that describes an interferometer operated in a regime in which device charging energy is small compared to the energy of formation of charged quasiparticles, which indicates that we have observed anyonic braiding. An interferometer device is used to detect the quantum-mechanical phase that is gained when two anyons are braided around each other. The fractional value of the phase proves that these quasiparticles are neither bosons nor fermions.