Complex Topology Research Articles

LS-category and topological complexity of Milnor manifolds and corresponding generalized projective product spaces

Milnor manifolds are a class of certain codimension-$1$ submanifolds of the product of projective spaces. In this paper, we study the LS-category and topological complexity of these manifolds. We determine the exact value of the LS-category and, in many cases, the topological complexity of these manifolds. We also obtain tight bounds on the topological complexity of these manifolds. It is known that Milnor manifolds admit $\mathbb{Z}_2$ and circle actions. We compute bounds on the equivariant LS-category and equivariant topological complexity of these manifolds. Finally, we describe the mod-$2$ cohomology rings of some generalized projective product spaces corresponding to Milnor manifolds and use this information to compute the bound on LS-category and topological complexity of these spaces.

An Alloy Engineering Strategy toward Helical Microstructures of Achiral π-Conjugated Molecules for Circularly Polarized Luminescence.

Helical assembly has been demonstrated to efficiently facilitate the circularly polarized luminescence (CPL) performances, but the synthesis of micro- and nanohelices from rigid achiral π-conjugated compounds remains challenging due to the absence of bilayer structures or complementary hydrogen-bonding interactions. Here, we develop an alloying strategy for the realization of helical microstructures from achiral anthracene/anthracene derivatives with x-/x-axis modification or anthracene/tetracene derivatives with x-/y-axis modification via solution coassembly. Interestingly, two anthracene derivatives bearing asymmetric phenyl/phenylethynyl groups and symmetric phenylethynyl groups can assemble into spiral microribbons with a fractal branching pattern. Using such an alloying strategy, color-tailorable ternary spiral microtubules/microribbons referring to high-efficiency energy transfer processes are achievable. Molecular dynamics simulations reveal that the Von Mises stress produced by symmetry differences of two components induces symmetry breaking of alloy structures associated with twisting. Additionally, the contents of the guest and H2O also play a vital role in the formation of intricate helical microstructures. Single binary and ternary spiral microribbons present considerable CPL properties with a dissymmetric factor ('glum') of more than 0.01. The present work provides new insights into the formation of helical microcrystals with complex topologies and new optoelectronic functions.

Topological complexity of finding flex points on cubic plane curves

We prove a lower bound for the topological complexity, in the sense of Smale, of the problem of finding a flex point on a cubic plane curve. The key is to bound the Schwarz genus of a cover associated to this problem. We also show that our lower bound for the complexity is close to the best possible.

Energy storages on the ferroelectric microstructures with transformation and nano vortex pattern

The energy storage and conversion in ferroelectrics can be realized through the microstructures of polar domains and domain walls, which resulting in the transformations from macro/microdomains to nanodomains or forming complex polar topologies. The physical basic models are adopted with domains and domain walls including 90o, 180o, 71o and 109o which are classified into two categories of 180o and α-angle, and are reconstructed with equivalent circuits simplified according to the reported patterns. Although electrical energy is known to be maintained by the charging capacitor, the energy storage effect on ferroelectric microstructure has been rarely explored for the relative paucity of experimental patterns reported with domains and domain walls. The diagrammatic sketches of transformation into nanodomain and vortex pattern are designed, and their respective formulas of total capacitances and energy densities are derived with crucial structural features. The findings reveal novel mechanisms of the relationship between energy storage and microstructures, that may be used to propose effective creation strategies or to design modern measure equipment in future.

On sequential versions of distributional topological complexity

On sequential versions of distributional topological complexity

Research on topology and control strategy of distribution network flexible interconnected device based on single-phase control unit and contact switch in series

There is an urgent need for flexible interconnection in the distribution network. However, due to the complex topology, control strategy and development cost of the existing flexible interconnected device (FID), the existing FID is difficult to be popularized and applied in the above scenarios. In view of the above problems, this paper constructs the FID topology based on the series connection of single-phase control unit and contact switch, and constructs the equivalent circuit of the interconnection line between FID and contact switch in series. The working principle of FID flexible closing loop and power flow control is studied from the dimension of FID output voltage and equivalent impedance. The FID flexible closing loop energy extraction strategy is proposed, and the FID system-level and device-level control strategies with flexible closing loop and power flow control functions are proposed. The simulation results show that the flexible interconnection device based on the proposed topology and control strategy can effectively compensate the pressure difference of the connecting points in the distribution network, regulate the power flow of the active distribution network, and realize the interconnection and mutual assistance of the distribution network.

Buckle-Barrel Correspondence Based on Topological Polarization Conversion in Mechanical Metamaterials.

Harnessing instabilities in architected lattices and metamaterials allows for controlled nonlinear deformations, enabling desired mechanical functions like shape morphing and energy absorption. By precisely tailoring instabilities such as buckling into the structure, deformations become a powerful instrument rather than a failure mode, offering new possibilities for predictable responses to mechanical loads. Inspired by the bulk-boundary correspondence in condensed matter physics, an analogous relationship is explored in metamaterials, where the underlying topology dictates whether the structure under load buckles or barrels. The underlying mechanism that originates from a polarization conversion in the elementary beam network is discussed. Moreover, these predictions, which can be extended to more complex topologies and higher dimensions, showcase that a mode inversion process governs both global deformations and the orientation of localized shear strains within the bulk. It is anticipated that the buckle-barrel correspondence can be extended to non-Hookean materials, paving the way for predicting the onset and evolution of truefailure.

DFT-calculations and RDG analysis (topology) of complexes between glutathione and nucleic bases of pyrymidine series.

Interactions between proteins and RNA, as well as between their structural fragments, are widespread in biological objects. We obtained the optimized structures of complexes of the glutathione anion with neutral molecules of uracil, thymine and cytosine. It was established that all complexes are stabilized by hydrogen bonds. The preference for various H-donors in nucleic base molecules (HN(1) or HN(3) in uracyl and thymine, N(1) or H2N in cytosine) for hydrogen bonding with the peptide has been analyzed. Chain elongation from dipeptide to tripeptide creates favorable conditions for increasing the number of hydrogen bonds in the complex. The strongest hydrogen bonds are formed with the carboxylate group of the peptide. Energy advantage of complexation with cytosine compared to other pyrimidine bases, and advantage of complexation with thymine compared to uracil have been established. The contributions of structural rearrangement of molecules, intermolecular interactions and H-bonding to the total values ​​of potential energy and Gibbs energy of the complexation process have been discussed. The article combines the results of calculations by the DFT/ B97D/6-311 + + G(3d,3p) and QTAIM methods to model the structure of ion-molecular complexes between the tripolar anion of peptide (glutathione) and neutral nucleic bases (uracil, thymine, cytosine). The PCM was used for solvent (water). Conformational analysis of the glutathione molecule was performed by scanning the potential energy while varying the dihedral angles. Several initial structure of peptide - nucleic base complexes with different modes of coordination were created in accordance with the MEP results. Non-covalent specific interactions in the complex were highlighted by RDG analysis. The hydrogen bond energies in complexes were calculated based on the correlation with the electron density at bond critical points. Changes in the total energy and Gibbs energy during complex formation, as well as contributions ​​from intermolecular interactions and structural rearrangement of reagent molecules, were determined.

Multi-material 3D printed composites inspired by nacre: a hard/soft mechanical interplay

Structural materials are used extensively in nature where mechanical function is required. These structures are composites consisting of soft and, in some cases, hard phases precisely distributed over different length scales. Bio-inspiration aims at producing materials with structure, design and/or mechanical properties adopted from biological tissues. To reproduce complex structures found in nature, additive manufacturing (AM) using three-dimensional printing (3DP) is an attractive method to assemble complex topologies with resolutions approaching the micro and nano-composition. Specifically, high-resolution MultiJetPrinting (MJP) 3D printing allows the simultaneous deposition of soft and hard photo curable plastic resins. Nacre is a prevalent example of a complex biological composite material organization that can test the ability of MJP to manufacture a bio-inspired engineering structure, where the organization of materials in nacre is optimized to avoid catastrophic failure. The ability to generate complex 3D organizations required to mimic the structure of nacre by controlled organization of soft and hard materials is achieved here using a generative design approach. Such a generative design is further enhanced by incorporating two differing MJP directions that provide relatively strong and weak interfaces between the soft and hard material phases. Consideration of classical stress transfer theory between at a hard material reinforcement interface was shown to correlate with experimental observations of mechanical performance and failure in 3D printed nacre inspired composites. Thus, the ability to distribute materials with a range of mechanical properties and incorporate further interfacial design is demonstrated. The approach presented is flexible and allows complex bio-inspired composites to be 3D printed that incorporate different interfacial quality through changing printing direction.

Toward the Evolutionary Optimisation of Small Molecules Within Coarse-Grained Simulations: Training Molecules to Hide Behind Lipid Head Groups.

Exploring the vast chemical space of small molecules poses a significant challenge. We develop a new strategy to efficiently explore this space using coarse-grained toy-like molecules utilizing the Martini3 force field and graph representations. This yields initial proof-of-concept results for the approach enabling the identification of optimal molecules with specific properties targeting lipid bilayers. By leveraging genetic algorithms and coarse-grained molecular dynamics simulations, we demonstrate the potential of our method in designing simple, linear molecules. Our findings show a good convergence toward molecules with weak amphiphilic properties, resembling known (general anesthetic) molecules. While this study demonstrates the feasibility of our method, further refinement is needed to fully realize its potential and explore more complex molecular topologies. Nevertheless, these encouraging results suggest a new path for future research in small molecule discovery and design without relying on extensive data sets.

Entanglement transition in random rod packings

Random packings of stiff rods are self-supporting mechanical structures stabilized by long-range interactions induced by contacts. To understand the geometrical and topological complexity of the packings, we first deploy X-ray computerized tomography to unveil the structure of the packing. This allows us to directly visualize the spatial variations in density, orientational order, and the entanglement, a mesoscopic field that we define in terms of a local average crossing number, a measure of the topological complexity of the packing. We find that increasing the aspect ratio of the constituent rods in a packing leads to a proliferation of regions of strong entanglement that eventually percolate through the system and correlated with a sharp transition in the mechanical stability of the packing. To corroborate our experimental findings, we use numerical simulations of contacting elastic rods and characterize their stability to static and dynamic loadings. Our experiments and computations lead us to an entanglement phase diagram which we also populate using published experimental data from pneumatically tangled filaments, worm blobs, and bird nests along with additional numerical simulations using these datasets. Together, these show the regimes associated with mechanically stable entanglement as a function of the statistics of the packings and loading, with lessons for a range of systems from reconfigurable architectures and textiles to active morphable filamentous assemblies.

Folding of a tandemly knotted protein: Evidence that a polypeptide chain can get out of deep kinetic traps.

It is hard to imagine how proteins can thread and form knots in their polypeptide chains, but they do. These topologically complex structures have challenged the traditional protein folding views of simple funnel-shaped energy landscapes. Previous experimental studies on the folding mechanisms of deeply knotted proteins with a single trefoil knot have yielded evidence that this topology has a more complicated folding landscape than other simpler proteins. However, to date, there have been no attempts to study the folding of any protein in which multiple threading events are needed to create more than one knot within a single polypeptide chain. Here, we report the construction and characterization of an artificial tandemly knotted protein. We find compelling evidence that both domains of the protein form trefoil knots with similar structures and stabilities to the parent single trefoil-knotted protein. In addition, we show that this tandemly knotted protein has a complex folding pathway in which there are additional very slow folding phases that we propose correspond to the formation of the second knot within the system. We also find evidence that during folding this protein gets transiently trapped in deep kinetic traps, however, the majority of protein chains (>90%) manage to partially unfold and acquire the native tandem-knot topology. This work highlights the fact that Nature can tolerate more complex protein topologies than we thought, and despite considerable misfolding during folding, protein chains can find their way to the native state even in the absence of molecular chaperones.

Deep Reinforcement Learning-Based Routing Method for Low Earth Orbit Mega-Constellation Satellite Networks with Service Function Constraints.

Low-orbit satellite communication networks have gradually become the research focus of fifth-generation (5G) beyond and sixth generation (6G) networks due to their advantages of wide coverage, large communication capacity, and low terrain influence. However, the low earth orbit mega satellite network (LEO-MSN) also has difficulty in constructing stable traffic transmission paths, network load imbalance and congestion due to the large scale of network nodes, a highly complex topology, and uneven distribution of traffic flow in time and space. In the service-based architecture proposed by 3GPP, the introduction of service function chain (SFC) constraints exacerbates these challenges. Therefore, in this paper, we propose GDRL-SFCR, an end-to-end routing decision method based on graph neural network (GNN) and deep reinforcement learning (DRL) which jointly optimize the end-to-end transmission delay and network load balancing under SFC constraints. Specifically, this method constructs the system model based on the latest NTN low-orbit satellite network end-to-end transmission architecture, taking into account the SFC constraints, transmission delays, and network node loads in the end-to-end traffic transmission, uses a GNN to extract node attributes and dynamic topology features, and uses the DRL method to design specific reward functions to train the model to learn routing policies that satisfy the SFC constraints. The simulation results demonstrate that, compared with graph theory-based methods and reinforcement learning-based methods, GDRL-SFCR can reduce the end-to-end traffic transmission delay by more than 11.3%, reduce the average network load by more than 14.1%, and increase the traffic access success rate and network capacity by more than 19.1% and two times, respectively.

Water intrusion in hydrophobic MOFs with complex topology: A glimpse of the intrusion mechanism of Cu2(tebpz).

Despite water intrusion in microporous materials being extensively investigated, obtaining a detailed overview of the intrusion mechanism in materials with more complex morphology, topology, and physical-chemical characteristics, such as metal-organic frameworks (MOFs), is far from trivial. In this work, we present a qualitative study on the mechanism of water intrusion in a crystallite of hydrophobic Cu2(tebpz) (tebpz = 3,3',5,5'-tetraethyl-4,4'-bipyrazolate) MOF. This MOF is characterized by a complex morphology; it consists of primary (main channels) and secondary (lateral apertures) porosities. This is similar to some zeolites, such as the so-called ITT-type zeolite framework, but it presents the additional characteristics of high flexibility of the material and non-uniform hydrophobicity. Interestingly, in Cu2(tebpz), water intrusion occurs first for some of the channels lying tangent to the surface of the MOF's crystallite. This is due to hydrogen bonding bridging with bulk water across the (thin) lateral apertures of these channels. In macroscopic terms, this can be understood as a local reduction of hydrophobicity favoring intrusion. Temperature and pressure influence the average number of hydrogen bonds and the number of intruded water molecules, explaining the effect of these thermodynamic parameters on the intrusion/extrusion characteristics of this porous material. Molecular dynamics simulations allowed us to glimpse liquid intrusion in this complex hydrophobic material, highlighting how the classical models valid for mesoporous systems, namely, Young-Laplace's law, are not quite appropriate to describe intrusion in such materials.

Hexagonal lattice diagrams for complex curves in ℂℙ²

We demonstrate that the geometric, topological, and combinatorial complexities of certain surfaces in C P 2 \mathbb {CP}^2 are closely related: We prove that a positive genus surface K \mathcal {K} in C P 2 \mathbb {CP}^2 that minimizes genus in its homology class is isotopic to a complex curve C d \mathcal {C}_d if and only if K \mathcal {K} admits a hexagonal lattice diagram, a special type of shadow diagram in which arcs meet only at bridge points and tile the central surface of the standard trisection of C P 2 \mathbb {CP}^2 by hexagons. There are eight families of these diagrams, two of which represent surfaces in efficient bridge position. Combined with a result of Lambert-Cole relating symplectic surfaces and bridge trisections, this allows us to provide a purely combinatorial reformulation of the symplectic isotopy problem in C P 2 \mathbb {CP}^2 . Finally, we show that that the varieties V d = { [ z 1 : z 2 : z 3 ] ∈ C P 2 : z 1 z 2 d − 1 + z 2 z 3 d − 1 + z 3 z 1 d − 1 = 0 } \mathcal {V}_d = \{[z_1:z_2:z_3] \in \mathbb {CP}^2 : z_1z_2^{d-1} + z_2z_3^{d-1} + z_3z_1^{d-1} = 0\} and V d ′ = { [ z 1 : z 2 : z 3 ] ∈ C P 2 : z 1 d − 1 z 2 + z 2 d − 1 z 3 + z 3 d − 1 z 1 = 0 } \mathcal {V}’_d = \{[z_1:z_2:z_3] \in \mathbb {CP}^2 : z_1^{d-1}z_2 + z_2^{d-1}z_3 + z_3^{d-1}z_1 = 0\} are in efficient bridge position with respect to the standard Stein trisection of C P 2 \mathbb {CP}^2 , and their shadow diagrams agree with the two families of efficient hexagonal lattice diagrams. As a corollary, we prove that two infinite families of complex hypersurfaces in C P 3 \mathbb {CP}^3 admit efficient Stein trisections, partially answering a question of Lambert-Cole and Meier.

A Conformal Transformation Technique for Mapping an Open Channel Fluid Flow into a Two-Dimensional Plane

Due to the emergence of many applications in areas of open-channel fluid flow, interest in this branch of fluid mechanics has increased considerably. These areas are wide-ranging, including electricity generation using tidal waves, control of floods and improvement of irrigation systems. Finance, thermal imaging, harnessing of melting of glaciers, flow of fluids over ramps and sluice gates, and quite notably, petroleum exploration are other areas where open channel flow mathematics find great relevance. Whether the free surface of the open channel is being predicted using known bottom characteristics (which is called the ‘direct approach’), or the bottom is being inferred from an observable surface (referred to as the ‘inverse approach’), researchers often find the need to make computations easier by transforming complex topologies into simpler, more relatable physical equivalents. In this paper, the channel flow of a gravity-influenced Newtonian fluid is treated as the physical plane. The fluid is flowing in the positive direction. The upper half-plane is conformally equivalent to the interior domain determined by any polygon, and the interior points of the physical plane are transformed to the corresponding points above the real axis of the upper half-plane which is then mapped onto the auxiliary half-plane, (the plane) by being treated as an infinite strip by use of the Schwarz-Christoffel theorem. Assumptions are made that the fluid has dimensional quantities such as uniform spee far upstream, and velocity potential . Far upstream before the arbitrary obstacle is encountered, the fluid has a uniform height, Then and are used to nondimensionalize the variables to enable computation in a completely non-dimensional environment. With the fluid assumed as steady, inviscid, irrotational and incompressible, w representing the physical w-plane, (and t) being points on the auxiliary the - plane, the angle made by a tangent to this plane at designated points, the required mapping is found to be .

Helical Polymer-Containing Bottlebrush Polymers (BBPs): Design, Synthesis, and Perspectives.

Helical polymer-containing bottlebrush polymers (BBPs) are a special and fascinating type of polymer. They possess bottlebrush topology and contain helical polymers as main chains (MCs) or side chains (SCs), thereby presenting interesting and fantastic properties, such as chiral amplification, circularly polarized luminescence, photonic crystal, and so on. This review mainly focuses on BBPs containing helical polymers of polypeptides, polyacetylenes (PAs), and polyisocyanides (PIs). Detailed summarizations are severally given to BBPs with helical polypeptides as MCs and SCs. Meanwhile, BBPs comprising helical PAs as MCs are fully discussed. What's more, BBPs consisted of helical PIs as MCs and SCs are described separately. In addition, BBPs with other helical polymers are briefly introduced, too. The authors hope this review will motivate more interest in developing helical polymers with complex topologies and fascinating properties, and encourage further progress in functional chiral materials.

Deep(er) reconstruction of imaging Cherenkov detectors with swin transformers and normalizing flow models

Abstract Imaging Cherenkov detectors are crucial for particle identification (PID) in nuclear and particle physics experiments. Fast reconstruction algorithms are essential for near real-time alignment, calibration, data quality control, and efficient analysis. At the future electron–ion collider (EIC), the ePIC detector will feature a dual Ring Imaging Cherenkov (RICH) detector in the hadron direction, a Detector of Internally Reflected Cherenkov (DIRC) in the barrel, and a proximity focus RICH in the electron direction. This paper focuses on the DIRC detector, which presents complex hit patterns and is also used for PID of pions and kaons in the experiment at JLab. We present Deep(er)RICH, an extension of the seminal DeepRICH work, offering improved and faster PID compared to traditional methods and, for the first time, fast and accurate simulation. This advancement addresses a major bottleneck in Cherenkov detector simulations involving photon tracking through complex optical elements. Our results leverage advancements in Vision Transformers, specifically hierarchical Swin Transformer and normalizing flows. These methods enable direct learning from real data and the reconstruction of complex topologies. We conclude by discussing the implications and future extensions of this work, which can offer capabilities for PID for multiple cutting-edge experiments like the future EIC.

Thermal emission modulation of fabrication-friendly, free-form metasurfaces via explainable deep-learning Bayesian optimization

Free-form metasurfaces with superimposed transformative meta-atoms provide a versatile platform to realize cross-band thermal emission control. However, design and manufacturing of free-form metasurfaces is extremely challenging, owing to the complex and fractal sub-wavelength topology. Here, we address these two issues by proposing an explainable deep-learning Bayesian optimization (DeepBO) framework to realize a library of fabrication-friendly, free-form metasurfaces with different light–matter interaction bandwidths. The DeepBO requires only 50 training data and is capable of screening high-dimensional design space of 1043 thermal photonic structure candidates with bandwidths from 0.3 to 3.2 eV. We unfold the black-box of deep-learning process by pattern recognition and identify the sub-space key features in the high-dimensional design space, which provides insights for thermal photonic metasurface design. We showcase the design and manufacturing of the broadband solar absorber and the narrowband thermophotovoltaic emitter with record-high spectral efficiency. The spectral selectivity of the fabricated free-form metasurface matches well with the design. The fabrication-friendly, free-form metasurfaces realized in this work can be generalized to thermal emitters for broad-ranges applications in energy and sensing.

Topological cascade of quantum Borromean rings

The evolution and the topological cascade of quantum vortices forming Borromean rings are studied for the first time. The initial configuration of the system is given by three elliptical planar loops linked together, and the evolution is governed by the numerical implementation of the Gross–Pitaevskii equation. It is found that the topological cascade is not unique, but it depends crucially on the initial geometric configuration. Quantum vortices undergo a series of spontaneous reconnections, resulting in various degenerative pathways characterized by different topology and structural complexity triggered by the different inclination of one of the initial ellipses. Typical decaying routes are given by the successive creation of a Whitehead link, a connected sum of two Hopf links, a trefoil knot, a Hopf link, and the final formation of unknotted, unlinked loops. By structural complexity analysis, we show that the generic trend of the vortex decay goes through a series of topological simplifications, resulting in the formation of small-scale planar loops (rings). During the later stage of evolution, the inverse cascade and topological cycles involving the interaction of unknotted loops become more common, remaining sub-dominant to the overall topological simplification process. These results pave the way to investigate the fundamental relations between structural complexity and energy contents.