International Journal of Topology

Schwarzites are negatively curved sp²-carbon frameworks that can be described as realizations of triply periodic minimal surface (TPMS) nets. This work examines topologically expanded Schwarzite networks derived from P-surface tilings by isolated heptagons, in which neighboring cages are connected via inserted carbon nanotubes rather than through direct links present in the parent Schwarzite. The study focuses on how such topological modifications influence network stability, density, and mechanical response. Density functional theory calculations show that nanotube-mediated expansion systematically reduces framework density while redistributing curvature within the network. A key topological distinction arises between structures formed by separating large cages and those formed by separating small cages. Separation of large cages is equivalent to inserting nanotube segments into adjacent small cages, increasing their effective size and relieving curvature-induced strain, thereby enhancing energetic stability. In contrast, separation of small cages preserves their topology and provides only limited strain relief. Cohesive energy trends correlate with the hexagon-to-heptagon ratio, approaching the graphene limit as topological expansion increases. Mechanical properties follow the same hierarchy, with strain-relieved networks displaying reduced stiffness and greater compliance. These findings indicate that the properties of Schwarzites are governed primarily by topological connectivity and curvature distribution rather than geometric scaling alone, establishing general principles for tuning stability and elasticity in negatively curved carbon networks through controlled topological expansion. Full article

In topological materials, dissipationless edge currents are protected against local defect scattering by the bulk inverted band structure and band gap. We propose that large uniaxial strain can effectively switch a 2D Chern insulator to a topologically trivial state. Further, we suggest that the boundary between strained and unstrained regions of a sample can act as a new edge for dissipationless current flow. Using density functional theory (DFT) calculations we demonstrate the strain-tunability of the monolayer ${\mathrm{MnBi}}_2{\mathrm{S}}_2{\mathrm{Te}}_2$ band structure and the switching of the Chern number. We combine uniaxial and biaxial strain results to map out the strain-tuned topological phase diagram. Full article

We introduce a notion of curvature based on informational holonomy. Let $(M,g)$ be a smooth Riemannian manifold and let $\pi :\mathcal{P}\to M$ be a bundle of state spaces equipped fibrewise with a smooth divergence $D_x$ inducing an information metric $g^{{\mathcal{P}}_x}$. Assuming a connection on $\mathcal{P}$ compatible with this fibrewise information geometry, we measure the deviation of holonomy around small geodesic triangles by transporting a reference state ${\mu }_x$ and comparing it to its image via the induced informational distance $d_x=\sqrt{2D_x}$. Normalizing the resulting distance defect by the geometric area yields a continuous informational holonomy (sectional) curvature$K_{\mathrm{hol}}^{\mathrm{cont}}(x,\mathsf{\Pi })$. We prove that this limit exists for all $(x,\mathsf{\Pi })$ and equals the norm of a vector $W_x(\mathsf{\Pi };{\mu }_x)\in T_{{\mu }_x}{\mathcal{P}}_x$ depending linearly on the curvature of the connection along $\mathsf{\Pi }$. In geometric models induced from the Levi–Civita connection via an isometric representation, $K_{\mathrm{hol}}^{\mathrm{cont}}$ becomes a scalar invariant of $R^g{|}_{\mathsf{\Pi }}$ and, on spaces of constant sectional curvature, reduces to a constant multiple of $|{\sec }_g|$. On the discrete side, we consider quasi-uniform sampling graphs whose edges carry channels approximating parallel transport. Discrete triangle holonomies define a curvature estimator, and under explicit sampling, area-approximation, and channel-consistency assumptions, we establish a discrete-to-continuum convergence theorem with a quantitative error bound controlled by the sampling scale. Full article

Many machine-learning tasks involve structured data whose geometry, local feature distributions, and global organization interact in ways that are not well captured by existing methods based on vectorization, graph metrics, or homological signatures. We introduce Fiber Bundle Learning (FBL), a topological framework that represents each data sample as a discrete fiber bundle and extracts a classification signature combining persistent homology, local feature geometry, and gluing structure. FBL builds a base space from the coarse geometry of each object, models local feature patches as fibers, and estimates transition maps between neighboring fibers to construct a discrete connection. From this representation, FBL computes a set of invariants: persistent homology of the base, fibers, and total space; holonomy obtained by transporting fiber states along cycles; curvature-like quantities measuring transition inconsistency; and discrete analogues of characteristic classes. These components are assembled into a fixed-length feature vector that can be used with any standard classifier. We show that FBL yields a signature with three desirable theoretical properties: stability under perturbations of geometry and local features, invariance under isometries and global fiber reparameterizations, and robustness to sampling noise. Our synthetic experiments show that FBL distinguishes twisted from untwisted bundles with identical homology, a distinction classical topological methods fail to capture. Additional tests quantify the system’s resistance to noise, its invariance to geometric transformations, and the contribution of each signature component. Taken together, our results indicate that representing data through fiber bundle structure may provide an effective tool for classifying complex, multi-level objects. Full article

We propose a topological data analysis framework for the study of damage evolution in anisotropic composite materials based on scalar filtrations defined on cubical complexes. Two complementary anisotropic filtrations are constructed from the structure tensor: a fibre-oriented filtration f1, capturing directional coherence, and a crack-oriented filtration f2, sensitive to isotropic and weakly oriented structures. Zero-dimensional persistent homology is analysed through merge trees built from the superlevel-set filtration via the transformation $g=1-f$, providing a hierarchical representation of connected components. Higher-order connectivity is described using skeleton-based Reeb-like graphs. From these constructions, we derive spatial and global descriptors, including a topological danger map and a Topological Damage Complexity Index (TDCI) based on one-dimensional persistent homology. The behaviour of the TDCI is examined with respect to variations in its parameters and to image perturbations, showing consistent trends across the considered configurations. The results highlight complementary structural behaviours captured by the two filtrations and show a coherent correspondence with observed patterns. Overall, the proposed framework provides a mathematically grounded description of structural organisation. It is intended as an exploratory approach, and further work is needed to clarify its relationship with the underlying physical damage mechanisms. Full article

Link congestion resulting from routine traffic patterns or malicious link flooding attacks (LFAs) poses a significant challenge in datacenter environments. The growing adoption of software-defined networking (SDN) offers a flexible framework for dynamic network reconfiguration, making it a promising approach for mitigating LFAs. Traffic redirection in SDN can follow either the shortest alternative path or a path that minimizes the number of rule modifications. The shortest alternative path sometimes yields a high number of changes in rules. As SDN switches are constrained by limited rule storage, a high number of rules may result in slow processing of the packet forwarding. Excessive rule updates can also degrade performance and introduce interruptions while the rules are being updated. This study focuses on minimizing rule changes when rerouting traffic away from congested links. We formulate two optimization problems aimed at reducing rule modifications during redirection. The first addresses a congested link and a specific flow, for which we propose solutions based on Dijkstra’s algorithm. The second extends to scenarios involving multiple congested links, introducing flow grouping and enhanced rule-merging strategies. To validate our approach, we conduct comprehensive simulations and real-world experiments within a datacenter setting. Full article

The advancement of autonomous robotic systems has led to significant capabilities in perception, localization, mapping, and control, yet a critical challenge remains in representing and preserving relational semantics, contextual reasoning, and cognitive transparency essential for collaboration in dynamic, human-centric environments. This paper introduces a unified architecture comprising the Ontology Neural Network (ONN) and the Ontological Real-Time Semantic Fabric (ORTSF) to address this challenge. The ONN formalizes relational semantic reasoning as a dynamic topological process by embedding Forman–Ricci curvature, persistent homology, and semantic tensor structures within a unified loss formulation, aiming to maintain relational integrity as scenes evolve. Building upon ONN, the ORTSF transforms reasoning traces into actionable control commands while compensating for system delays through predictive operators designed to preserve phase margins. Theoretical analysis and extensive simulations demonstrate that ORTSF maintains designed phase margins, offering advantages over classical delay compensation methods. Empirical studies indicate the framework’s effectiveness in unifying semantic cognition and robust control, providing a mathematically principled solution for cognitive robotics. Full article

We develop a topological classification of admissible reconstruction operations in generative systems where extended structure is built through repeated local extension subject to compatibility constraints. Reconstruction is formalized as a feasibility-governed process rather than a dynamical or metric one, with admissibility determined by the accumulation of obstruction under composition. Using loop diagnostics, we identify global incompatibilities that are invisible to local extension rules but become unavoidable under closed composition. Under mild and realization-independent assumptions, including indefinite continuation and finite interface capacity, we show that persistent nontrivial obstruction is possible only when it is supported on codimension-2 subsets of the reconstructed domain. This result induces a small number of topological universality classes distinguished by the existence and stability of loop-detectable obstruction. The framework is model-agnostic and applies equally to discrete, combinatorial, and continuum reconstructions, providing a topological explanation for the ubiquity of codimension-2 defects in generative systems. Full article

We study Geraghty-type non-self mappings within the framework of best proximity point theory. By introducing auxiliary functions with subsequential convergence, we establish general conditions ensuring the existence and uniqueness of best proximity points. Our results extend and unify earlier work on proximal and Kannan-type contractions under a Geraghty setting, and provide counterexamples showing that the auxiliary assumptions are essential. As an illustration, we construct an explicit non-self alignment mapping on subsets of ${\mathbb{R}}^2$ for which all hypotheses can be verified and the unique best proximity point, as well as the convergence of the associated proximal iteration, can be computed in closed form. Full article

Vertical wind shear plays a crucial role in the organization and persistence of mesoscale convective systems, yet its geometrical and topological effects remain challenging to quantify. In this study, we introduce a shear-induced anisotropic metric, denoted $d_S$, which embeds the direction and magnitude of environmental wind shear directly into the framework of persistent homology. The metric deforms the ambient geometry by weighting distances differently along and across the shear direction, enabling topological descriptors to respond dynamically to the flow environment. We establish the analytical properties of $d_S$, and demonstrate its compatibility with Vietoris–Rips filtrations. The method is applied to the Corsican bow–echo event of 18 August 2022, where shear vectors are derived from ERA5 reanalysis data. Two complementary topological analyses are performed: a transport analysis on $H_0$ using Wasserstein distances, and a structural analysis on $H_1$ persistent generators under parallel and perpendicular shear metrics. The results reveal distinct topological evolutions associated with different shear orientations, highlighting the sensitivity of persistent homology to shear-induced deformation. Overall, the framework provides a mathematically consistent bridge between dynamical meteorology and topological data analysis, extending persistent homology to anisotropic metric spaces. Full article

In this paper we present an implementation of a computer algorithm that automatically determines the topological structure of spacetime, using a branched covering space representation. This algorithm is applied to a few simple examples in dimension 3, and a complete set of the fundamental groups realized over several graphs is found. We also include some new visualizations of the branched covering construction, in order to aid and clarify the understanding of how these structures can be used in quantum gravity to realize the topological nature of the spacetime foam. Full article

Let $\eta :X\to \mathbb{R}$ be a Morse function on a connected closed manifold X. We denote by $C\left(\eta \right)$ the fiber product of two copies of $\eta$. For Morse functions $f:M\to \mathbb{R}$ and $g:N\to \mathbb{R}$, we define the function $f\ast g:M\times N\to \mathbb{R}$ by $(f\ast g)(p,q):=f\left(p\right)·g\left(q\right)$. The purpose of this paper is twofold: Firstly, we study the sufficient condition for which $\chi \left(C\right(f\ast g\left)\right)=\chi \left(C\right(f\left)\right)·\chi \left(C\right(g\left)\right)$ holds, where $\chi$ denotes the Euler characteristic. Secondly, for the case that f is the well-known Morse function on $\mathbb{C}{P}^n$, we determine $\chi \left(C\right(f\ast f\left)\right)$. Full article

This paper investigates quantum contextuality, a central nonclassical aspect of quantum mechanics, by employing the algebraic and topological structures of modular tensor categories. The analysis establishes that braid group representations constructed from modular categories, including the $SU{\left(2\right)}_k$ and Fibonacci anyon models, inherently produce state-dependent contextuality, as revealed by measurable violations of noncontextuality inequalities. The explicit construction of unitary representations on fusion spaces allows this paper to identify a direct structural correspondence between braiding operations and logical contextuality frameworks. The results offer a comprehensive topological framework to classify and quantify contextuality in low-dimensional quantum systems, thereby elucidating its role as a resource in topological quantum computation and advancing the interface between quantum algebra, topology, and quantum foundations. Full article

We present a comprehensive investigation into the emergence of interface-bound states, particularly Majorana zero modes (MZMs), in a lateral heterostructure composed of two three-dimensional topological insulators (TIs), Bi₂Se₃ and Sb₂Te₃, under the influence of proximity-induced superconductivity from niobium (Nb) contacts. We develop an advanced two-dimensional Dirac model for the topological surface states (TSS), incorporating spatially varying chemical potentials and s-wave superconducting pairing. Using the Bogoliubov–de Gennes (BdG) formalism, we derive analytical solutions for the bound states and compute the local density of states (LDOS) at the interface, revealing zero-energy modes characteristic of MZMs. The topological nature of these states is rigorously analyzed through winding numbers and Pfaffian invariants, and their robustness is explored under various physical perturbations, including gating effects. Our findings highlight the potential of this heterostructure as a platform for topological quantum computing, with detailed predictions for experimental signatures via tunneling spectroscopy. Full article

Microtubules are cylindrical protein polymers that organize the cytoskeleton and play essential roles in intracellular transport, cell division, and possibly cognition. Their highly ordered, quasi-crystalline lattice of tubulin dimers, notably tryptophan residues, endows them with a rich topological and arithmetic structure, making them natural candidates for supporting coherent excitations at optical and terahertz frequencies. The Penrose–Hameroff Orch OR theory proposes that such coherences could couple to gravitationally induced state reduction, forming the quantum substrate of conscious events. Although controversial, recent analyses of dipolar coupling, stochastic resonance, and structured noise in biological media suggest that microtubular assemblies may indeed host transient quantum correlations that persist over biologically relevant timescales. In this work, we build upon two complementary approaches: the parametric resonance model of Nishiyama et al. and our arithmetic–geometric framework, both recently developed in Quantum Reports. We unify these perspectives by describing microtubules as rectangular lattices governed by the imaginary quadratic field $\mathbb{Q}\left(i\right)$, within which nonlinear dipolar oscillations undergo stochastic parametric amplification. Quantization of the resonant modes follows Gaussian norms $N=p^2+q^2$, linking the optical and geometric properties of microtubules to the arithmetic structure of $\mathbb{Q}\left(i\right)$. We further connect these discrete resonances to the derivative of the elliptic L-function, $L\prime (E,1)$, which acts as an arithmetic free energy and defines the scaling between modular invariants and measurable biological ratios. In the appended adelic extension, this framework is shown to merge naturally with the Bost–Connes and Connes–Marcolli systems, where the norm character on the ideles couples to the Hecke character of an elliptic curve to form a unified adelic partition function. The resulting arithmetic–elliptic resonance model provides a coherent bridge between number theory, topological quantum phases, and biological structure, suggesting that consciousness, as envisioned in the Orch OR theory, may emerge from resonant processes organized by deep arithmetic symmetries of space, time, and matter. Full article

The field of probabilistic metric spaces has an intrinsic interest based on a blend of ideas drawn from metric space theory and probability theory. The goal of the present paper is to introduce and study new ideas in this field. In general terms, we investigate the following concepts: linearly ordered families of distances and associated continuity properties, geometric properties of distances, finite range weak probabilistic metric spaces, generalized Menger spaces, and a categorical framework for weak probabilistic metric spaces. Hopefully, the results will contribute to the foundations of the subject. Full article

We prove that in every nonempty perfect Polish space, every dense $G_{\delta }$ subset contains strictly decreasing and strictly increasing chains of dense $G_{\delta }$ subsets of length $\mathfrak{c}$, the cardinality of the continuum. As a corollary, this holds in ${\mathbb{R}}^n$ for each $n\ge 1$. This provides an easy answer to a question of Erdős since the set of Liouville numbers admits a descending chain of cardinality $\mathfrak{c}$, each member of which has the Erdős property. We also present counterexamples demonstrating that the result fails if either the perfection or the Polishness assumption is omitted. Finally, we show that the set $\mathcal{T}$ of real Mahler T-numbers is a dense Borel set and contains a strictly descending chain of length $\mathfrak{c}$ of proper dense Borel subsets. Full article

We propose a geometry topological framework to analyze storm dynamics by coupling persistent homology with Anti-de Sitter (AdS)-inspired metrics. On radar images of a bow echo event, we compare Euclidean distance with three compressive AdS metrics ($\alpha$ = 0.01, 0.1, 0.3) via time-resolved $H_1$ persistence diagrams for the arc and its internal cells. The moderate curvature setting ($\alpha =0.1$) offers the best trade-off: it suppresses spurious cycles, preserves salient features, and stabilizes lifetime distributions. Consistently, the arc exhibits longer, more dispersed cycles (large-scale organizer), while cells show shorter, localized patterns (confined convection). Cross-correlations of $H_1$ lifetimes reveal a temporal asymmetry: arc activation precedes cell activation. A differential indicator $\Delta \left(t\right)$ based on Wasserstein distances quantifies this divergence and aligns with the visual onset in radar, suggesting early warning potential. Results are demonstrated on a rapid Corsica bow echo; broader validation remains future work. Full article

This article develops an interdisciplinary framework that applies topological and graph-theoretical methods to public procurement markets and digital platform economies. Conceptualizing legal–economic interactions as dynamic networks of nodes and edges, we show how structural properties—centrality, clustering, connectivity, and boundary formation—shape contestability, resilience, and compliance. Using EU-relevant contexts (public procurement directives and the Digital Markets Act), we formalize network representations for buyers, suppliers, platforms, and regulators; define operational indicators; and illustrate an empirical, value-weighted buyer → supplier network to reveal a sparse but highly modular architecture with a high-value backbone. We then map these structural signatures to concrete legal levers (lotting and framework design, modification scrutiny, interoperability and data-access duties) and propose dashboard-style diagnostics for proactive oversight. The findings demonstrate how topological modelling complements doctrinal analysis by making hidden architectures visible and by linking measurable structure to regulatory outcomes. We conclude with implications for evidence-informed regulatory design and a research agenda integrating graph analytics, comparative evaluation across jurisdictions, and machine-learning-assisted anomaly detection. Full article