Search Results (25)

This article examines quantum group symmetry using the Potts model. The transformation of the Potts model into a polynomial knot state on Kaufman square brackets is analyzed. It is shown how a dichromatic polynomial for a planar graph can be obtained using Temperley–Lieb operator algebra. The proposed work provides insight into the $7_4$ knot-partition function of Takara Musubi using a strain factor that represents the particles in the lattice knots of the above-mentioned model. As far as theoretical physics is concerned, this statement provides a correct explanation of the connection between the Potts model and the similar square lattice of knot and link invariants. Full article

Although electrons (fermions)and photons (bosons) produce the same interference patterns in the two-slit experiments, known in optics for photons since the 17th Century, the description of these patterns for electrons and photons thus far was markedly different. Photons are spin one, relativistic and massless particles while electrons are spin half massive particles producing the same interference patterns irrespective to their speed. Experiments with other massive particles demonstrate the same kind of interference patterns. In spite of these differences, in the early 1930s of the 20th Century, the isomorphism between the source-free Maxwell and Dirac equations was established. In this work, we were permitted replace the Born probabilistic interpretation of quantum mechanics with the optical. In 1925, Rainich combined source-free Maxwell equations with Einstein’s equations for gravity. His results were rediscovered in the late 1950s by Misner and Wheeler, who introduced the word "geometrodynamics” as a description of the unified field theory of gravity and electromagnetism. An absence of sources remained a problem in this unified theory until Ranada’s work of the late 1980s. However, his results required the existence of null electromagnetic fields. These were absent in Rainich–Misner–Wheeler’s geometrodynamics. They were added to it in the 1960s by Geroch. Ranada’s solutions of source-free Maxwell’s equations came out as knots and links. In this work, we establish that, due to their topology, these knots/links acquire masses and charges. They live on the Dupin cyclides—the invariants of Lie sphere geometry. Symmetries of Minkowski space-time also belong to this geometry. Using these symmetries, Varlamov recently demonstrated group-theoretically that the experimentally known mass spectrum for all mesons and baryons is obtainable with one formula, containing electron mass as an input. In this work, using some facts from polymer physics and differential geometry, a new proof of the knotty nature of the electron is established. The obtained result perfectly blends with the description of a rotating and charged black hole. Full article

The Kronecker algebra $\mathcal{K}$ is the path algebra induced by the quiver with two parallel arrows, one source and one sink (i.e., a quiver with two vertices and two arrows going in the same direction). Modules over $\mathcal{K}$ are said to be Kronecker modules. The classification of these modules can be obtained by solving a well-known tame matrix problem. Such a classification deals with solving systems of differential equations of the form $Ax=B{x}^{\prime }$, where A and B are $m\times n$, $\mathbb{F}$-matrices with $\mathbb{F}$ an algebraically closed field. On the other hand, researching the Yang–Baxter equation (YBE) is a topic of great interest in several science fields. It has allowed advances in physics, knot theory, quantum computing, cryptography, quantum groups, non-associative algebras, Hopf algebras, etc. It is worth noting that giving a complete classification of the YBE solutions is still an open problem. This paper proves that some indecomposable modules over $\mathcal{K}$ called pre-injective Kronecker modules give rise to some algebraic structures called skew braces which allow the solutions of the YBE. Since preprojective Kronecker modules categorize some integer sequences via some appropriated snake graphs, we prove that such modules are automatic and that they induce the automatic sequences of continued fractions. Full article

Currently, researching the Yang–Baxter equation (YBE) is a subject of great interest among scientists of diverse areas in mathematics and other sciences. One of the fundamental open problems is to find all of its solutions. The investigation deals with developing theories such as knot theory, Hopf algebras, quandles, Lie and Jordan (super) algebras, and quantum computing. One of the most successful techniques to obtain solutions of the YBE was given by Rump, who introduced an algebraic structure called the brace, which allows giving non-degenerate involutive set-theoretical solutions. This paper introduces Brauer configuration algebras, which, after appropriate specializations, give rise to braces associated with Thompson’s group F. The dimensions of these algebras and their centers are also given. Full article

We recently proposed that topological quantum computing might be based on $SL(2,\mathbb{C})$ representations of the fundamental group ${\pi }_1(S^3\backslash K)$ for the complement of a link K in the three-sphere. The restriction to links whose associated $SL(2,\mathbb{C})$ character variety $\mathcal{V}$ contains a Fricke surface ${\kappa }_d=xyz-x^2-y^2-z^2+d$ is desirable due to the connection of Fricke spaces to elementary topology. Taking K as the Hopf link $L2a1$, one of the three arithmetic two-bridge links (the Whitehead link $5_1^2$, the Berge link $6_2^2$ or the double-eight link $6_3^2$) or the link $7_3^2$, the $\mathcal{V}$ for those links contains the reducible component ${\kappa }_4$, the so-called Cayley cubic. In addition, the $\mathcal{V}$ for the latter two links contains the irreducible component ${\kappa }_3$, or ${\kappa }_2$, respectively. Taking $\rho$ to be a representation with character ${\kappa }_d$ ($d<4$), with $\left|x\right|,\left|y\right|,\left|z\right|\le 2$, then $\rho \left({\pi }_1\right)$ fixes a unique point in the hyperbolic space ${\mathcal{H}}_3$ and is a conjugate to a $SU\left(2\right)$ representation (a qubit). Even though details on the physical implementation remain open, more generally, we show that topological quantum computing may be developed from the point of view of three-bridge links, the topology of the four-punctured sphere and Painlevé VI equation. The 0-surgery on the three circles of the Borromean rings L6a4 is taken as an example. Full article

Taking a DNA sequence, a word with letters/bases A, T, G and C, as the relation between the generators of an infinite group $\pi$, one can discriminate between two important families: (i) the cardinality structure for conjugacy classes of subgroups of $\pi$ is that of a free group on one to four bases, and the DNA word, viewed as a substitution sequence, is aperiodic; (ii) the cardinality structure for conjugacy classes of subgroups of $\pi$ is not that of a free group, the sequence is generally not aperiodic and topological properties of $\pi$ have to be determined differently. The two cases rely on DNA conformations such as A-DNA, B-DNA, Z-DNA, G-quadruplexes, etc. We found a few salient results: Z-DNA, when involved in transcription, replication and regulation in a healthy situation, implies (i). The sequence of telomeric repeats comprising three distinct bases most of the time satisfies (i). For two-base sequences in the free case (i) or non-free case (ii), the topology of $\pi$ may be found in terms of the $SL(2,\mathbb{C})$ character variety of $\pi$ and the attached algebraic surfaces. The linking of two unknotted curves—the Hopf link—may occur in the topology of $\pi$ in cases of biological importance, in telomeres, G-quadruplexes, hairpins and junctions, a feature that we already found in the context of models of topological quantum computing. For three- and four-base sequences, other knotting configurations are noticed and a building block of the topology is the four-punctured sphere. Our methods have the potential to discriminate between potential diseases associated to the sequences. Full article

Understanding the photophysical properties and stability of near-infrared fluorescent proteins (NIR FPs) based on bacterial phytochromes is of great importance for the design of efficient fluorescent probes for use in cells and in vivo. Previously, the natural ligand of NIR FPs biliverdin (BV) has been revealed to be capable of covalent binding to the inherent cysteine residue in the PAS domain (Cys15), and to the cysteine residue introduced into the GAF domain (Cys256), as well as simultaneously with these two residues. Here, based on the spectroscopic analysis of several NIR FPs with both cysteine residues in PAS and GAF domains, we show that the covalent binding of BV simultaneously with two domains is the reason for the higher quantum yield of BV fluorescence in these proteins as a result of rigid fixation of the chromophore in their chromophore-binding pocket. We demonstrate that since the attachment sites are located in different regions of the polypeptide chain forming a figure-of-eight knot, their binding to BV leads to shielding of many sites of proteolytic degradation due to additional stabilization of the entire protein structure. This makes NIR FPs with both cysteine residues in PAS and GAF domains less susceptible to cleavage by intracellular proteases. Full article

It is shown that the representation theory of some finitely presented groups thanks to their $S{L}_2\left(\mathbb{C}\right)$ character variety is related to algebraic surfaces. We make use of the Enriques–Kodaira classification of algebraic surfaces and related topological tools to make such surfaces explicit. We study the connection of $S{L}_2\left(\mathbb{C}\right)$ character varieties to topological quantum computing (TQC) as an alternative to the concept of anyons. The Hopf link H, whose character variety is a Del Pezzo surface $f_H$ (the trace of the commutator), is the kernel of our view of TQC. Qutrit and two-qubit magic state computing, derived from the trefoil knot in our previous work, may be seen as TQC from the Hopf link. The character variety of some two-generator Bianchi groups, as well as that of the fundamental group for the singular fibers ${\tilde{E}}_6$ and ${\tilde{D}}_4$ contain $f_H$. A surface birationally equivalent to a $K_3$ surface is another compound of their character varieties. Full article

Symmetry (and group theory) is a fundamental principle of theoretical physics. Finite symmetries, continuous symmetries of compact groups, and infinite-dimensional representations of noncompact Lie groups are at the core of solid physics, particle physics, and quantum physics, respectively. The latter groups now play an important role in many branches of mathematics. In more recent years, we have been faced with the impact of topological quantum field theory (TQFT). Topology and symmetry have deep connections, but topology is inherently broader and more complex. While the presence of symmetry in physical phenomena imposes strong constraints, topology seems to be related to low-energy states and is very likely to provide information about the different dynamical trajectories and patterns that particles can follow. For example, regarding the relationship of topology to low-energy states, Hodge’s theory of harmonic forms shows that the zero-energy states (for differential forms) correspond to the cohomology. Regarding the relationship of topology to particle trajectories, a topological knot can be seen as an orbit with complex properties in spacetime. The various deformations or embeddings of the knot, performed in low or high dimensions, allow defining different equivalence classes or topological types, and interestingly, it is possible from these types to study the symmetries associated with the deformations and their changes. More specifically, in the present work, we address two issues: first, that quantum geometry deforms classical geometry, and that this topological deformation may produce physical effects that are specific to the quantum physics scale; and second, that mirror symmetry and the phenomenon of topological change are closely related. This paper was aimed at understanding the conceptual and physical significance of this connection. Full article

A highly enantioselective protocol has been recently described as allowing the synthesis of five-membered cyclic imines harnessing the selective generation of a β-Csp³-centered radical of acyl heterocyclic derivatives and its subsequent interaction with diverse NH-ketimines. The overall transformation represents a novel cascade process strategy crafted by individual well-known steps; however, the construction of the new C-C bond highlights a crucial knot from a mechanistically perspective. We believe that the full understanding of this enigmatic step may enrich the current literature and expand latent future ideas. Therefore, a detailed mechanistic study of the protocol has been conducted. Here, we provide theoretical insight into the mechanism using quantum chemistry calculations. Two possible pathways have been investigated: (a) imine reduction followed by radical–radical coupling and (b) radical addition followed by product reduction. In addition, investigations to unveil the origin behind the enantioselectivity of the 1-pyrroline derivatives have been conducted as well. Full article

Hydrogen peroxide (H₂O₂) is a broad-range chemical catalyst that is receiving rapidly increasing attention recently due to its role as a signaling molecule in various plant physiological and biochemical processes. A study was carried out to investigate the effects of H₂O₂ on the plant physiology, root growth, mineral nutrient accumulation, root anatomy, and nematode control of Ficus deltoidea, a slow growing shade tolerant and nematode susceptible medicinal plant. H₂O₂ at 0 (control), 15, 30, 60, and 90 mM was injected into the root zone of plants weekly. The results showed that the treatment of H₂O₂ enhanced the accumulation of pigments, photosynthetic characteristics, and quantum yield (F_v/F_m) of F. deltoidea. H₂O₂ at a 90 mM treatment significantly increased seedling height, leaf number, syconium number, biomass yield, relative water content, leaf dry matter, leaf moisture, and live line fuel moisture of the plant by 1.35-, 3.02-, 3.60-, 5.13-, 1.21-, 1.12-, 1.79- and 1.06-fold, respectively, over the control plant. In addition, root growth, which includes root crown diameter, root length, root volume, root tips, number of roots and root biomass, also exhibited the highest values with an application of 90 mM of H₂O₂. Heavy metals arsenic (As⁺) and antimony (Sb⁺) content in the leaves decreased by 4.08-and 1.63-fold, respectively, in the 60 mM H₂O₂ treated plant when compared to the control plant. In addition, 90 mM H₂O₂ was the best treatment for magnesium (Mg²⁺), calcium (Ca²⁺), and sodium (Na⁺) mineral accumulation in the syconium of F. deltoidea. Treatments with 60 mM H₂O₂ increased magnesium (Mg²⁺), calcium (Ca²⁺), and potassium (K⁺) content in leaves by 14%, 19%, and 15%, respectively, over the control plant. In the study of controlling root-knot nematode, both control and 15 mM treatments produced many root galls, whereas, 60 mM H₂O₂ treatment produced fewer tiny root galls and 90 mM of H₂O₂ showed no root gall formation. H₂O₂ treatments reduced root gall size, root/shoot ratio, and increased the shoot biomass of plants. The treated root developed an epidermal suberin, root periderm, resin duct, cortex, druses, and a well-developed vascular system compared to the control plants. Furthermore, no nematodes were observed in the roots of treated plants with 30–90 mM H₂O₂. The study concluded that injections of 60–90 mM H₂O₂ to the root zone weekly improved plant physiology, increased mineral accumulation, root growth and development, reduced root gall formation, improved root cellular structure, and controlled root-knot nematode of F. deltoidea plants. Full article

What is the origin of quantum randomness? Why does the deterministic, unitary time development in Hilbert space (the ‘$4\pi$-realm’) lead to a probabilistic behaviour of observables in space-time (the ‘$2\pi$-realm’)? We propose a simple topological model for quantum randomness. Following Kauffmann, we elaborate the mathematical structures that follow from a $distinction$$(A,B)$ using group theory and topology. Crucially, the 2:1-mapping from $SL(2,\mathbb{C})$ to the Lorentz group $SO(3,1)$ turns out to be responsible for the stochastic nature of observables in quantum physics, as this 2:1-mapping breaks down during interactions. Entanglement leads to a change of topology, such that a distinction between A and B becomes impossible. In this sense, entanglement is the counterpart of a distinction $(A,B)$. While the mathematical formalism involved in our argument based on virtual Dehn twists and torus splitting is non-trivial, the resulting haptic model is so simple that we think it might be suitable for undergraduate courses and maybe even for High school classes. Full article

In this paper, we will present some ideas to use 3D topology for quantum computing. Topological quantum computing in the usual sense works with an encoding of information as knotted quantum states of topological phases of matter, thus being locked into topology to prevent decay. Today, the basic structure is a 2D system to realize anyons with braiding operations. From the topological point of view, we have to deal with surface topology. However, usual materials are 3D objects. Possible topologies for these objects can be more complex than surfaces. From the topological point of view, Thurston’s geometrization theorem gives the main description of 3-dimensional manifolds. Here, complements of knots do play a prominent role and are in principle the main parts to understand 3-manifold topology. For that purpose, we will construct a quantum system on the complements of a knot in the 3-sphere. The whole system depends strongly on the topology of this complement, which is determined by non-contractible, closed curves. Every curve gives a contribution to the quantum states by a phase (Berry phase). Therefore, the quantum states can be manipulated by using the knot group (fundamental group of the knot complement). The universality of these operations was already showed by M. Planat et al. Full article

We argue that the usual Bloch sphere is insufficient in various aspects for the representation of qubits in quantum information theory. For example, spin flip operations with the quaternions $IJK=e^{\frac{2\pi i}{2}}=-1$ and $JIK=+1$ cannot be distinguished on the Bloch sphere. We show that a simple knot theoretic extension of the Bloch sphere representation is sufficient to track all unitary operations for single qubits. Next, we extend the Bloch sphere representation to entangled states using knot theory. As applications, we first discuss contextuality in quantum physics—in particular the Kochen-Specker theorem. Finally, we discuss some arguments against many-worlds theories within our knot theoretic model of entanglement. The key ingredients of our approach are symmetries and geometric properties of the unitary group. Full article

The concepts of tile number and space-efficiency for knot mosaics were first explored by Heap and Knowles in 2018, where they determined the possible tile numbers and space-efficient layouts for every prime knot with mosaic number 6 or less. In this paper, we extend those results to prime knots with mosaic number 7. Specifically, we find the possible values for the number of non-blank tiles used in a space-efficient $7\times 7$ mosaic of a prime knot are 27, 29, 31, 32, 34, 36, 37, 39, and 41. We also provide the possible layouts for the mosaics that lead to these values. Finally, we determine which prime knots can be placed within the first of these layouts, resulting in a list of knots with mosaic number 7 and tile number 27. Full article