Topological Knot Research Articles

Optical induction of monopoles, knots, and skyrmions in quantum gases

Single topological defects and related structures have been experimentally created inside clouds of Bose-Einstein condensates. However, a practical protocol for creating multiple defects inside a single cloud—a prerequisite for studying defect interactions—has been lacking. Here we propose, and theoretically analyze, a practical protocol for the creation of configurations of topological monopoles, quantum knots, and skyrmions in Bose-Einstein condensates by employing fictitious magnetic fields induced by the interaction of the atomic cloud with coherent light fields. It is observed that a single coherent field is not enough for this purpose, but instead we find incoherent superpositions of several coherent fields that introduce topological point charges. We numerically estimate the experimentally achievable strengths and gradients of the induced fictitious magnetic fields and find them to be adjustable at will to several orders of magnitude greater than those of the physical magnetic fields employed in previous experimental studies. This property together with ultrafast control of the optical fields paves the way for advanced engineering of topological defects in quantum gases. Published by the American Physical Society 2024

An example of a non-extendable isotopy of a knot

The paper constructs an example of a (non-ambient) isotopy of a (topological) knot in [Formula: see text] which does not extend to an isotopy of a two-component link whose second component is a meridianal ring of this knot. By constructing the example, the paper answers a question raised in 2005 by Melikhov. Apart from presenting the corresponding example, the paper will discuss the consequences that the discovery of this example has for attacking a problem that was in literature asked in 1974 by Rolfsen: Namely deciding, whether there exist knots (and it is clear that at most everywhere wild knots might have such a property) that even with respect to ordinary (not necessarily ambient) isotopy are non-equivalent to the trivial knot.

Knotted artifacts in predicted 3D RNA structures.

Unlike proteins, RNAs deposited in the Protein Data Bank do not contain topological knots. Recently, admittedly, the first trefoil knot and some lasso-type conformations have been found in experimental RNA structures, but these are still exceptional cases. Meanwhile, algorithms predicting 3D RNA models have happened to form knotted structures not so rarely. Interestingly, machine learning-based predictors seem to be more prone to generate knotted RNA folds than traditional methods. A similar situation is observed for the entanglements of structural elements. In this paper, we analyze all models submitted to the CASP15 competition in the 3D RNA structure prediction category. We show what types of topological knots and structure element entanglements appear in the submitted models and highlight what methods are behind the generation of such conformations. We also study the structural aspect of susceptibility to entanglement. We suggest that predictors take care of an evaluation of RNA models to avoid publishing structures with artifacts, such as unusual entanglements, that result from hallucinations of predictive algorithms.

Tying a true topological protein knot by cyclization

Knotted proteins are fascinating to biophysicists because of their robust ability to fold into intricately defined three-dimensional structures with complex and topologically knotted arrangements. Exploring the biophysical properties of the knotted proteins is of significant interest, as they could offer enhanced chemical, thermal, and mechanostabilities. A true mathematical knot requires a closed path; in contrast, knotted protein structures have open N- and C-termini. To address the question of how a truly knotted protein differs from the naturally occurring counterpart, we enzymatically cyclized a 31 knotted YibK protein from Haemophilus influenza (HiYibK) to investigate the impact of path closure on its structure-function relationship and folding stability. Through the use of a multitude of structural and biophysical tools, including X-ray crystallography, NMR spectroscopy, small angle X-ray scattering, differential scanning calorimetry, and isothermal calorimetry, we showed that the path closure minimally perturbs the native structure and ligand binding of HiYibK. Nevertheless, the cyclization did alter the folding stability and mechanism according to chemical and thermal unfolding analysis. These molecular insights contribute to our fundamental understanding of protein folding and knotting that could have implications in the protein design with higher stabilities.

Geometric learning of knot topology.

Knots are deeply entangled with every branch of science. One of the biggest open challenges in knot theory is to formalise a knot invariant that can unambiguously and efficiently distinguish any two knotted curves. Additionally, the conjecture that the geometrical embedding of a curve encodes information on its underlying topology is, albeit physically intuitive, far from proven. Here we attempt to tackle both these outstanding challenges by proposing a neural network (NN) approach that takes as input a geometric representation of a knotted curve and tries to make predictions of the curve's topology. Intriguingly, we discover that NNs trained with a so-called geometrical "local writhe" representation of a knot can distinguish curves that share one or many topological invariants and knot polynomials, such as mutant and composite knots, and can thus classify knotted curves more precisely than some knot polynomials. Additionally, we also show that our approach can be scaled up to classify all prime knots up to 10-crossings with more than 95% accuracy. Finally, we show that our NNs can also be trained to solve knot localisation problems on open and closed curves. Our main discovery is that the pattern of "local writhe" is a potentially unique geometric signature of the underlying topology of a curve. We hope that our results will suggest new methods for quantifying generic entanglements in soft matter and even inform new topological invariants.

Structure, dynamics, and stability of the smallest and most complex 71 protein knot

Proteins can spontaneously tie a variety of intricate topological knots through twisting and threading of the polypeptide chains. Recently developed artificial intelligence algorithms have predicted several new classes of topological knotted proteins, but the predictions remain to be authenticated experimentally. Here, we showed by X-ray crystallography and solution-state NMR spectroscopy that Q9PR55, an 89-residue protein from Ureaplasma urealyticum, possesses a novel 71 knotted topology that is accurately predicted by AlphaFold 2, except for the flexible N terminus. Q9PR55 is monomeric in solution, making it the smallest and most complex knotted protein known to date. In addition to its exceptional chemical stability against urea-induced unfolding, Q9PR55 is remarkably robust to resist the mechanical unfolding-coupled proteolysis by a bacterial proteasome, ClpXP. Our results suggest that the mechanical resistance against pulling-induced unfolding is determined by the complexity of the knotted topology rather than the size of the molecule.

An entanglement constraint model of topological knot in highly entangled gel towards ultra-high toughness

Highly entangled gels have gained extensive attention due to their excitingly large deformation and high toughness. To understand the toughening mechanism of these highly entangled gels, an entanglement constraint model has been established, based on the spatially prismatic constraint and Gaussian distribution models. A free-energy function is formulated to study the conformational dynamics, rubbery elasticity and sliding effect of topological knots in the entangled chains. Monte Carlo, molecular dynamics and finite element analysis were conducted to verify the coupling effect of inter-chain entanglement and intra-chain knot topology on the toughness behavior of highly entangled gels. Finally, experimental data available in the literature were used to verify the proposed models, providing a physical insight into the toughening mechanism of inter-chain entanglement constraint and intra-chain knot topology in the highly entangled gel.

Computational Exploration of Multistable Elastic Knots

We present an algorithmic approach to discover, study, and design multistable elastic knots. Elastic knots are physical realizations of closed curves embedded in 3-space. When endowed with the material thickness and bending resistance of a physical wire, these knots settle into equilibrium states that balance the forces induced by elastic deformation and self-contacts of the wire. In general, elastic knots can have many distinct equilibrium states, i.e. they are multistable mechanical systems. We propose a computational pipeline that combines randomized spatial sampling and physics simulation to efficiently find stable equilibrium states of elastic knots. Leveraging results from knot theory, we run our pipeline on thousands of different topological knot types to create an extensive data set of multistable knots. By applying a series of filters to this data, we discover new transformable knots with interesting geometric and physical properties. A further analysis across knot types reveals geometric and topological patterns, yielding constructive principles that generalize beyond the currently tabulated knot types. We show how multistable elastic knots can be used to design novel deployable structures and engaging recreational puzzles. Several physical prototypes at different scales highlight these applications and validate our simulation.

Lipschitz geometry of surface germs in {mathbb {R}}^4: metric knots

A link at the origin of an isolated singularity of a two-dimensional semialgebraic surface in {mathbb {R}}^4 is a topological knot (or link) in S^3. We study the connection between the ambient Lipschitz geometry of semialgebraic surface germs in {mathbb {R}}^4 and knot theory. Namely, for any knot K, we construct a surface X_K in {mathbb {R}}^4 such that: the link at the origin of X_{K} is a trivial knot; the germs X_K are outer bi-Lipschitz equivalent for all K; two germs X_{K} and X_{K'} are ambient semialgebraic bi-Lipschitz equivalent only if the knots K and K' are isotopic. We show that the Jones polynomial can be used to recognize ambient bi-Lipschitz non-equivalent surface germs in {mathbb {R}}^4, even when they are topologically trivial and outer bi-Lipschitz equivalent.

Uncolorable Brunnian Links are Linked

Summary The topology of knots and links can be studied by examining colorings of their diagrams. We explain how to detect knots and links using the method of Fox tricoloring, and we give a new and elementary proof that an infinite family of Brunnian links are each linked. Our proof is based on the remarkable fact (which we prove) that if a link diagram cannot be tricolored then it must be linked. Our paper introduces readers to the Fox coloring generalization of tricoloring and the further algebraic generalization, called a quandle by David Joyce.

On a generalization of Jones polynomial and its categorification for Legendrian knots

In this article, we explore a polynomial invariant for Legendrian knots which is a natural extension of Jones polynomial for (topological) knots. To this end, a new type of skein relation is introduced for the front projections of Legendrian knots. Further, we give a categorification of the polynomial invariant for Legendrian knots which is a natural extension of Khovanov homology. The Thurston-Bennequin invariant of Legendrian knot appears naturally in the construction of the homology as the grade shift.The constructions of the polynomial invariant and its categorification are natural in the sense that if we treat Legendrian knots as only knots (that is, we forget the geometry on the knots), then we recover the Jones polynomial and Khovanov homology respectively. In the end, we discuss strengths and limitations of these invariants.

Strain imposed by knotting

Modulating properties by regulating the topology of molecular knots has had a considerable impact on the development of molecular nanotopology. In a recent issue of <i>Nature Synthesis</i>, Liu and co-workers report the use of a "nested contra-helices" approach for generating a series of molecular trefoil knots with varied degrees of tightness.

High capacity topological coding based on nested vortex knots and links

Optical knots and links have attracted great attention because of their exotic topological characteristics. Recent investigations have shown that the information encoding based on optical knots could possess robust features against external perturbations. However, as a superior coding scheme, it is also necessary to achieve a high capacity, which is hard to be fulfilled by existing knot-carriers owing to the limit number of associated topological invariants. Thus, how to realize the knot-based information coding with a high capacity is a key problem to be solved. Here, we create a type of nested vortex knot, and show that it can be used to fulfill the robust information coding with a high capacity assisted by a large number of intrinsic topological invariants. In experiments, we design and fabricate metasurface holograms to generate light fields sustaining different kinds of nested vortex links. Furthermore, we verify the feasibility of the high-capacity coding scheme based on those topological optical knots. Our work opens another way to realize the robust and high-capacity optical coding, which may have useful impacts on the field of information transfer and storage.

Symmetry and Symmetry Breaking in Physics: From Geometry to Topology

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.

Topological Knots in Quantum Spin SystemsSupported by the National Natural Science Foundation of China (Grants Nos. 11874225, 11975128, and 11605094).

Knot theory provides a powerful tool for understanding topological matters in biology, chemistry, and physics. Here knot theory is introduced to describe topological phases in a quantum spin system. Exactly solvable models with long-range interactions are investigated, and Majorana modes of the quantum spin system are mapped into different knots and links. The topological properties of ground states of the spin system are visualized and characterized using crossing and linking numbers, which capture the geometric topologies of knots and links. The interactivity of energy bands is highlighted. In gapped phases, eigenstate curves are tangled and braided around each other, forming links. In gapless phases, the tangled eigenstate curves may form knots. Our findings provide an alternative understanding of phases in the quantum spin system, and provide insights into one-dimension topological phases of matter.

Converging experimental and computational views of the knotting mechanism of a small knotted protein

MJ0366 from Methanocaldococcus jannaschii is the smallest topologically knotted protein known to date. 92 residues in length, MJ0366 ties a trefoil (31) knot by threading its C-terminal helix through a buttonhole formed by the remainder of the secondary structure elements. By generating a library of point mutations at positions pertinent to the knot formation, we systematically evaluated the contributions of individual residues to the folding stability and kinetics of MJ0366. The experimental Φ-values were used as restraints to computationally generate an ensemble of conformations that correspond to the transition state of MJ0366, which revealed several nonnative contacts. The importance of these nonnative contacts in stabilizing the transition state of MJ0366 was confirmed by a second round of mutagenesis, which also established the pivotal role of F15 in stapling the network of hydrophobic interactions around the threading C-terminal helix. Our converging experimental and computational results show that, despite the small size, the transition state of MJ0366 is formed at a very late stage of the folding reaction coordinate, following a polarized pathway. Eventually, the formation of extensive native contacts, as well as a number of nonnative ones, leads to the threading of the C-terminal helix that defines the topological knot.

Construction of Unknotted and Knotted Symmetric Developable Bands

We describe a method for constructing developable bands with N ≥ 3 half twists. Each band is formed by threading a flat rectangular strip through a scaffold made from identical circular cylinders and smoothly connecting its short ends. The N cylinders in a scaffold are arranged with N-fold rotational symmetry. The number of half twists in a band is equal to the number N of cylinders in its scaffold and each band inherits the symmetry of its scaffold. Each scaffold admits a family of bands of the same length but variable width up to a maximum value determined by the features of the scaffold. Apart from orientable and nonorientable unknots, our method allows for the construction of bands with the topology of torus knots. We detail the geometric properties of the construction, discuss certain fundamental restrictions that must be met to ensure constructability, and calculate the elastic bending energy of each band. The rotational symmetry underlying the construction is essential for obtaining the presented bands, as the general non-symmetric problem is even more complex and has not yet been investigated. The bands and their corresponding scaffolds can be used as structural elements in practical applications, one of which we describe and analyze. The construction serves as a basis for a general framework for building a large variety of scaffolds and the corresponding unstretchable bands. Together, these assemblies can be used in architectural, interior, and machine design. They also open new avenues for the layout of conveyor belts in factories, airports, and other settings.

From the Lagrange polygon to the figure eight I

The present work studies the continuation class of the regular n-gon solution of the n-body problem. For odd numbers of bodies between $$n = 3$$ and $$n = 15$$ , we apply one parameter numerical continuation algorithms to the energy/frequency variable and find that the figure eight choreography can be reached starting from the regular n-gon. The continuation leaves the plane of the n-gon and passes through families of spatial choreographies with the topology of torus knots. Numerical continuation out of the n-gon solution is complicated by the fact that the kernel of the linearization there is high dimensional. Our work exploits a symmetrized version of the problem which admits dense sets of choreography solutions and which can be written as a delay differential equation in terms of one of the bodies. This symmetrized setup simplifies the problem in several ways. On the one hand, the direction of the kernel is determined automatically by the symmetry. On the other hand, the set of possible bifurcations is reduced and the n-gon continues to the eight after a single symmetry breaking bifurcation. Based on the calculations presented here, we conjecture that the n-gon and the eight are in the same continuation class for all odd numbers of bodies.

On the support genus of Legendrian knots

In this paper, we show that any topological knot or link in $$S^1 \times S^2$$ sits on a planar page of an open book decomposition whose monodromy is a product of positive Dehn twists. As a consequence, any knot or link type in $$S^1 \times S^2$$ has a Legendrian representative having support genus zero. We also show this holds for some knots and links in the lens spaces L(p, 1).

Structure and Oxidative Folding of AAI, the Major Alfa-Amylase Inhibitor From Amaranth Seeds.

AAI, the major alpha-amylase inhibitor (AAI) present in the seeds of the Mexican crop plant Amaranthus hypocondriacus is a 32-residue-long polypeptide with three disulfide bridges. Its structure is most closely related to the plant amylase inhibitor subfamily of knottins characterized by a topological knot formed by one disulfide bridge threading through a loop formed by the peptide chain as well as a short three-stranded beta sandwich core. AAI is specific against insect amylases and does not act on corresponding human or mammalian enzymes. It was found that the oxidative folding of AAI seems to follow a hirudine-like pathway with many non-native intermediates, but notably it proceeds through a major folding intermediate (MFI) that contains a vicinal disulfide bridge. Based on a review of the pertinent literature, the known vicinal disulfides in native proteins as well as well as the network of disulfide interchanges, we propose that MFI is a kinetic trap corresponding to a compact molten globule-like state which constrains the peptide chain to a smaller number of conformations that in turn can be rapidly funneled toward the native state.