International Journal of Topology

In this study of the Reidemeister moves within the classical knot theory, we focus on hard diagrams of knots and links, categorizing them as either rigid or shaky based on their adaptability to certain moves. We establish that every link possesses a diagram that is a rigid hard diagram, and we provide an upper limit for the number of crossings in such diagrams. Furthermore, we investigate rigid hard diagrams for specific knots or links to determine their rigid hard index. In the topic of shaky hard diagrams, we demonstrate the existence of such diagrams for the unknot and unlink of any number of components and present examples of shaky hard diagrams. Full article

This article proposes a unified concept of topological types of convergence for nets of multifunctions between topological spaces. Any kind of convergence is representable by a (2n + 2)-tuple, n = 0, 1, …, of two special functions u and l, such that their compositions $ul$ and $lu$ create the Choquet supremum and infimum operations, respectively, on the filters considered in terms of the upper Vietoris topology on the range hyperspace of the considered multifunctions. Convergence operators are defined by establishing the order of composition of the functions from such (2n + 2) tuples. An allocation of places for the two distinguished functions in a convergence operator reflects the structure of the used (2n + 2)-tuple. A monoid of special three-parameter functions called products describes the set of all possible structures. The monoid of products is the domain space of the convergence operators. The family of all convergence operators forms a finite monoid whose neutral element determines the pointwise convergence and possesses the structure determined by the neutral element of the monoid of products. We demonstrate the construction process of every convergence operator and show that the notions of the presented concept can characterize many well-known classical types of convergence. Of particular importance are the types of convergence derived from the concept of continuous convergence. We establish some general theorems about the necessary and sufficient conditions for the continuity of the limit multifunctions without any assumptions about the type of continuity of the members of the nets. Full article

The origin of glass formation has been one of the greatest mysteries of science. The first clues emerged in Ge_xSe_1-x glasses, where the bond-stretching and bond angle-bending constraints are countable, and it was found that the most favorable compositions for glass formation involved matching constraints with the degrees of freedom. Modulated-Differential Scanning Calorimetric (MDSC) studies on Ge_xSe_1-x chalcogenide glasses revealed two elastic phase transitions—a stiffness transition at x = 0.20 and a stress transition at x = 0.26—leading to the observation of three topological phases: a flexible phase at x < 0.20, an intermediate phase in the 0.20 < x < 0.26 range, and a stressed–rigid phase for compositions x > 0.26. The three topological phases (TPs) have now been generically observed in more than two dozen chalcogenides and modified oxide glasses. In proteins, the transition from the unfolded (flexible) to the folded (isostatically rigid intermediate) phase represents the stiffness transition. Self-organization causes proteins to display a dynamic reversibility of the folding process. The evolutions of protein dynamics may also exhibit stiffness phase transitions similar to those seen in glasses. Full article

Though a convergence space is connected if and only if its topological modification is connected, connected subsets of a convergence space differ from those of its topological modification. We explore which subsets exhibit connectedness for the convergence or for the topological modification. In particular, we show that connectedness of a subset is equivalent for a convergence or for its reciprocal modification and that the largest set enclosing a given connected subset of a convergence space is the adherence of the connected set for the reciprocal modification of the convergence. Full article

We introduce the notion of bornological approach nearness as a unified extension of various classical nearness structures. By redefining completeness within this framework, we establish a generalized version of the Niemytzki–Tychonoff theorem. Our results not only extend known compactness criteria in nearness spaces but also offer a new perspective that incorporates boundedness and bornological methods in the theory of approach spaces. Full article

Graph theory models relationships by representing entities as vertices and their interactionsas edges. To handle directionality and multiple head–tail assignments, various extensions—directed, bidirected, and multidirected graphs—have been introduced, with the multidirected graph unifying the first two. In this work, we further enrich this landscape by proposing the Multidirected hypergraph, which merges the flexibility of hypergraphs and superhypergraphs to describe higher-order and hierarchical connections. Building on this, we introduce five uncertainty-aware Multidirected frameworks—fuzzy, neutrosophic, plithogenic, rough, and soft multidirected graphs—by embedding classical uncertainty models into the Multidirected setting. We outline their formal definitions, examine key structural properties, and illustrate each with examples, thereby laying groundwork for future advances in uncertain graph analysis and decision-making. Full article

Classical algebraic structures—such as magmas, groups, and Lie groups—are characterized by increasingly strong requirements in binary operation, ranging from no additional constraints to associativity, identity, inverses, and smooth-manifold structures. The hyperstructure paradigm extends these notions by allowing the operation to return subsets of elements, giving rise to hypermagmas, hypergroups, and Lie hypergroups, along with their variants such as quotient, reduced, and fuzzy hypergroups. In this work, we introduce the concept of superhyperstructures, obtained by iterating the powerset construction, and develop the theory of superhypermagmas and Lie superhypergroups. We further define and analyze quotient superhypergroups, reduced superhypergroups, and fuzzy superhypergroups, exploring their algebraic properties and interrelationships. Full article

Let Y be a smooth compact n-manifold. We studied smooth embeddings and immersions $\beta :M\to \mathbb{R}\times Y$ of compact n-manifolds M such that $\beta \left(M\right)$ avoids some priory chosen closed poset $\mathrm{\Theta }$ of tangent patterns to the fibers of the obvious projection $\pi :\mathbb{R}\times Y\to Y$. Then, for a fixed Y, we introduced an equivalence relation between such $\beta$’s; creating a crossover between pseudo-isotopies and bordisms. We called this relation quasitopy. In the presented study of quasitopies, the spaces ${\mathcal{P}}_d^{\mathbf{c}\mathrm{\Theta }}$ of real univariate polynomials of degree d with real divisors, whose combinatorial patterns avoid a given closed poset $\mathrm{\Theta }$, play the classical role of Grassmanians. We computed the quasitopy classes ${\mathcal{Q}}_d^{\mathsf{emb}}(Y,\mathbf{c}\mathrm{\Theta })$ of $\mathrm{\Theta }$-constrained embeddings $\beta$ in terms of homotopy/homology theory of spaces Y and ${\mathcal{P}}_d^{\mathbf{c}\mathrm{\Theta }}$. We proved also that the quasitopies of embeddings stabilize, as $d\to \infty$. Full article

Persistent homology is a powerful tool in topological data analysis that captures the multi-scale topological features of data. In this work, we provide a mathematical introduction to persistent homology and demonstrate its application to protein–protein interaction networks. We combine persistent homology with algebraic connectivity, a graph-theoretic measure of network robustness, to analyze the topology and stability of PPI networks. An example is provided to illustrate the methodology and its potential applications in systems biology. Full article

A multicomplex structure is defined from an ordered lattice of multigraphs. This structure will help us to observe the features of persistent homology in this context, its interaction with the ordering, and the repercussions of merging multigraphs in the calculation of Betti numbers. For the latter, an extended version of the incremental algorithm is provided. The ideas developed here are mainly oriented to the original example described by the author and others in the context of the formalization of the notion of embodiment in Neuroscience. Full article

Raynaud’s Phenomenon (RP), characterized by episodic reductions in peripheral blood flow, leads to significant discomfort and functional impairment. Existing therapeutic strategies focus on pharmacological treatments, external heat supplementation and exercise-based rehabilitation, but fail to address biomechanical contributions to vascular dysfunction. We introduce a computational approach rooted in topological transformations of hand prehension, hypothesizing that specific hand postures can generate transient geometric structures that enhance thermal and hemodynamic properties. We examine whether a flexed hand posture—where fingers are brought together to form a closed-loop toroidal shape—may modify heat transfer patterns and blood microcirculation. Using a combination of heat diffusion equations, fluid dynamics models and topological transformations, we implement a heat transfer and blood flow simulation to examine the differential thermodynamic behavior of the open and closed hand postures. We show that the closed-hand posture may preserve significantly more heat than the open-hand posture, reducing temperature loss by an average of 1.1 ± 0.3 °C compared to 3.2 ± 0.5 °C in the open-hand condition (p < 0.01). Microvascular circulation is also enhanced, with a 53% increase in blood flow in the closed-hand configuration (p < 0.01). Therefore, our findings support the hypothesis that maintaining a closed-hand posture may help mitigate RP symptoms by preserving warmth, reducing cold-induced vasoconstriction and optimizing peripheral flow. Overall, our topologically framed approach provides quantitative evidence that postural modifications may influence peripheral vascular function through biomechanical and thermodynamic mechanisms, elucidating how shape-induced transformations may affect physiological and pathological dynamics. Full article

We establish a comprehensive framework demonstrating that physical reality can be understood as a holographic encoding of underlying computational structures. Our central thesis is that different geometric realizations of the same physical system represent equivalent holographic encodings of a unique computational structure. We formalize quantum complexity as a physical observable, establish its mathematical properties, and demonstrate its correspondence with geometric descriptions. This framework naturally generalizes holographic principles beyond AdS/CFT correspondence, with direct applications to black hole physics and quantum information theory. We derive specific, quantifiable predictions with numerical estimates for experimental verification. Our results suggest that computational structure, rather than geometry, may be the more fundamental concept in physics. Full article

We find that the whole set of known baryons of spin parity $J^P={{\displaystyle \frac{1}{2}}}^{+}$ (the ground state) and $J^P={{\displaystyle \frac{3}{2}}}^{+}$ (the first excited state) is organized in multiplets which may efficiently be encoded by the multiplets of conjugacy classes in the small finite group $G=(192, 187)$. A subset of the theory is the small group $(48, 29)\cong GL(2, 3)$ whose conjugacy classes are in correspondence with the baryon families of Gell-Mann’s octet and decuplet. G has many of its irreducible characters that are minimal and informationally complete quantum measurements that we assign to the baryon families. Since G is isomorphic to the group of braiding matrices of $SU{\left(2\right)}_2$ Ising anyons, we explore the view that baryonic matter has a topological origin. We are interested in the holographic gravity dual $Ad{S}_3/QF{T}_2$ of the Ising model. This dual corresponds to a strongly coupled pure Einstein gravity with central charge $c=1/2$ and AdS radius of the order of the Planck scale. Some physical issues related to our approach are discussed. Full article

One classic result in domain theory is that the Scott space of every domain (continuous directed complete poset) is sober. Johnstone constructed the first directed complete poset (dcpo for short) whose Scott space is not sober. This non-sober dcpo has been used in many other parts of domain theory and more properties of it have been uncovered. In this survey paper, we first collect and prove the major properties (some of which are new as far as we know) of Johnstone’s dcpo. We then propose a general method of constructing a dcpo from given posets and prove some properties. Some problems are posed for further investigation. This paper can serve as a relatively complete resource on Johnstone’s dcpo. Full article

This paper presents a novel approach to solving the Yang–Mills existence and mass gap problem using quantum information theory. We develop a rigorous mathematical framework that reformulates the Yang–Mills theory in terms of quantum circuits and entanglement structures. Our method provides a concrete realization of the Yang–Mills theory that is manifestly gauge-invariant and satisfies the Wightman axioms. We demonstrate the existence of a mass gap by analyzing the entanglement spectrum of the vacuum state, establishing a direct connection between the mass gap and the minimum non-zero eigenvalue of the entanglement Hamiltonian. Our approach also offers new insights into non-perturbative phenomena such as confinement and asymptotic freedom. We introduce new mathematical tools, including entanglement renormalization for gauge theories and quantum circuit complexity measures for quantum fields. The implications of our work extend beyond the Yang–Mills theory, suggesting new approaches to quantum gravity, strongly coupled systems, and cosmological problems. This quantum information perspective on gauge theories opens up exciting new directions for research at the intersection of quantum field theory, quantum gravity, and quantum computation. Full article

The twisted reflection operators are defined on the hyperbolic plane. They are then specialized in hyperbolic reflections, according to which the desymmetrized $PSL (2,\mathbb{Z})$ group is rewritten. The Bogomolny–Carioli transfer operators are newly analytically expressed in terms of the Dehn twists. The Bogomolny–Gauss mapping class group of the desymmetrized $PSL (2,\mathbb{Z})$ domain is newly proven. The paradigm to apply the Hecke theory on the CAT spaces on which the Dehn twists act is newly established. The Bogomolny–Gauss map is proven to be one of infinite topological entropy. Full article

The sheaf cohomology techniques are newly used to include Morse simplicial complexes in a rectangular-matrix chain, whose singular values are compatible with those of a square matrix, which can be used for multiple sequencing. The equivalence with the simplices of the corresponding graph is proven, as well as that the filtration of the corresponding probability space. The new protocol eliminates the problem of stochastic stability of deep Markov models. The paradigm can be implemented to develop deep-machine-learning multiple sequencing. The construction of the deep Markov models for sequencing, starting from a profile Markov model, is analytically written. Applications can be found as an amino-acid sequencing model. As a result, the nucleotide-dependence of the positions on the alignments are fully modelized. The metrics of the manifolds are discussed. The instance of the application of the new paradigm to the Jukes–Cantor model is successfully controlled on nucleotide-substitution models. Full article

This paper explores the application of Hilbert topological spaces and linear operator algebra in the modelling and analysis of OFDM signals and wireless channels, where the channel is considered as a linear time-invariant (LTI) system. The wireless channel, when subjected to an input OFDM signal, can be described as a mapping from an input Hilbert space to an output Hilbert space, with the system response governed by linear operator theory. By employing the mathematical framework of Hilbert spaces, we formalise the representation of OFDM signals, which are interpreted as elements of an infinite-dimensional vector space endowed with an inner product. The LTI wireless channel is characterised by using bounded linear operators on these spaces, allowing for the decomposition of complex channel behaviour into a series of linear transformations. The channel’s impulse response is treated as a kernel operator, facilitating a functional analysis approach to understanding the signal transmission process. This representation enables a more profound understanding of channel effects, such as fading and interference, through the eigenfunction expansion of the operator, leading to a spectral characterization of the channel. The algebraic properties of linear operators are leveraged to develop optimal solutions for mitigating channel distortion effects. Full article

Some polynomials find their way into chemical graph theory less often than others. They could provide new ways of understanding the origins of regularities in the chemistry of specific classes of compounds. This study’s objective is to depict the place of polynomials in chemical graph theory. Different approaches and notations are explained and levelled. The mathematical aspects of a series of such polynomials are put into the context of recent research. The directions in which this project was intended to proceed and where it stands right now are presented. Full article