Int. J. Topol., Volume 3, Issue 3 (September 2026) – 3 articles

We establish a metrization result for some $b_v\left(s\right)$-metric spaces, extending a result recently established for rectangular b-metric spaces. Furthermore, we introduce the notion of an extended polygonal b-metric space (or $b_v\left(\theta \right)$-metric space), which unifies and generalizes several classes of spaces, including metric spaces, rectangular metric spaces, b-metric spaces, rectangular b-metric spaces, polygonal metric spaces, and $b_v\left(s\right)$-metric spaces. Some fixed-point results in $b_v\left(\theta \right)$-metric spaces are established under the weak orbital completeness condition in the framework of the Banach contraction principle and for generalized expansive Hardy-Rogers-type mappings. An a priori error estimate for the iterative process is obtained in both $b_v\left(\theta \right)$-metric and $b_v\left(s\right)$-metric spaces. We also establish the Ulam-Hyers stability of fixed-point equations in both $b_v\left(\theta \right)$-metric and $b_v\left(s\right)$-metric spaces. Several examples are provided, and applications to certain types of integral equations and initial value problems are presented, illustrating the applicability and effectiveness of the obtained results. Full article

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