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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.

Three-dimensional braiding composite material has the advantages of high strength, high modulus, high temperature resistance, non-layered structure and easy design. Traditional high-performance metal materials will be replaced by 3D braiding composite material, so it has been highly valued and applied in many fields such as aerospace, weaponry and equipment. At present, there are many researches on 3D braiding technology, such as 3D rotary braiding machine and Cartesian 3D braiding machine in terms of braiding equipment. The research on the structure of preforms and the performance of composite materials are also more in-depth. The development of 3D braiding technology from the aspects of 3D fabric, braiding technology, equipment are summarized in this article. The development of 3D braiding technology requires high-speed automation and commercialization, this will be a challenge for 3D braiding technology in the future.

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.

To address the design challenges of braiding paths for complex cross-sectional preforms, this study develops a carrier path generation algorithm for a stepwise rotary braid 3D braiding machine incorporating graph theory. By applying graph theory, by converting the perform cross-sectional shape and the braiding chassis into graphs, a graph model transformation algorithm is established to optimize carrier paths and improve braiding efficiency. This algorithm considers the motion states of the driving components and the relative positions between carriers, converting complex braiding structures into optimal carrier paths. The effectiveness of the proposed algorithm was verified through three different carrier path planning case studies including square, rotational and equal-length multi-arm configurations, confirmed the methodology’s capability to maintain structural accuracy while minimizing deviations from design specifications. The results demonstrate that this method has significant potential in the automated arrangement of braiding trajectories for 3D braided complex cross-section components, laying the foundation for the further development of 3D braiding technology automation.

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°.

Non-Abelian braiding has attracted substantial attention because of its pivotal role in describing the exchange behaviour of anyons—candidates for realizing quantum logics. The input and outcome of non-Abelian braiding are connected by a unitary matrix that can also physically emerge as a geometric-phase matrix in classical systems. Hence it is predicted that non-Abelian braiding should have analogues in photonics, although a feasible platform and the experimental realization remain out of reach. Here we propose and experimentally realize an on-chip photonic system that achieves the non-Abelian braiding of up to five photonic modes. The braiding is realized by controlling the multi-mode geometric-phase matrix in judiciously designed photonic waveguide arrays. The quintessential effect of braiding—sequence-dependent swapping of photon dwell sites—is observed in both classical-light and single-photon experiments. Our photonic chips are a versatile and expandable platform for studying non-Abelian physics, and we expect the results to motivate next-generation non-Abelian photonic devices.Non-Abelian braiding—a candidate for realizing quantum logics—is demonstrated by controlling the geometric-phase matrix in a photonic chip, and its key characteristics are observed.

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.

Generalizing work of Marin [12], we construct in a unified way all the \"braids and ties'' algebras available in literature and new ones.

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 derive an explicit formula for the holonomy \\(R\\)-matrix of quantum \\(sl_2\\) at a root of unity. We show it factorizes into a product of four quantum dilogarithms and satisfies a holonomy Yang-Baxter equation. This factorization extends previously known results and we collect many existing results needed for our computation.