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Results for “knot theory”

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At least 19 records

All sources NASA NTRS DOE OSTI

DOE OSTI2022

Observation of a linked-loop quantum state in a topological magnet

Quantum phases can be classified by topological invariants, which take on discrete values capturing global information about the quantum state. Over the past decades, these invariants have come to play a central role in describing matter, providing the foundation for understanding superfluids, magnets, the quantum Hall effect, topological insulators, Weyl semimetals and other phenomena. Here we report an unusual linking-number (knot theory) invariant associated with loops of electronic band crossings in a mirror-symmetric ferromagnet. Using state-of-the-art spectroscopic methods, we directly observe three intertwined degeneracy loops in the material's three-torus, T 3 , bulk Brillouin zone. We find that each loop links each other loop twice. Through systematic spectroscopic investigation of this linked-loop quantum state, we explicitly draw its link diagram and conclude, in analogy with knot theory, that it exhibits the linking number (2, 2, 2), providing a direct determination of the invariant structure from the experimental data. Here we further predict and observe, on the surface of our samples, Seifert boundary states protected by the bulk linked loops, suggestive of a remarkable Seifert bulk-boundary correspondence. Our observation of a quantum loop link motivates the application of knot theory to the exploration of magnetic and superconducting quantum matter.

36 MATERIALS SCIENCE↗

DOE OSTI2021

Learning to Unknot

We introduce natural language processing into the study of knot theory, as made natural by the braid word representation of knots. We study the UNKNOT problem of determining whether or not a given knot is the unknot. After describing an algorithm to randomly generate N-crossing braids and their knot closures and discussing the induced prior on the distribution of knots, we apply binary classification to the UNKNOT decision problem. We find that the Reformer and shared-QK Transformer network architectures outperform fully-connected networks, though all perform at 95% accuracy. Perhaps surprisingly, we find that accuracy increases with the length of the braid word, and that the networks learn a direct correlation between the confidence of their predictions and the degree of the Jones polynomial. Finally, we utilize reinforcement learning (RL) to find sequences of Markov moves and braid relations that simplify knots and can identify unknots by explicitly giving the sequence of unknotting actions. Trust region policy optimization (TRPO) performs consistently well, reducing 80% of the unknots with up to 96 crossings we tested to the empty braid word, and thoroughly outperformed other RL algorithms and random walkers. Studying these actions, we find that braid relations are more useful in simplifying to the unknot than one of the Markov moves.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2025

Fusion and fission of particle-like chiral nematic vortex knots

Vortex knots have been seen decaying in many physical systems. Here we describe topologically protected vortex knots, which remain stable and undergo fusion and fission and conserve a topological invariant. The host medium, a chiral nematic liquid crystal, exhibits intrinsic chirality of molecular alignment, whereas cores of the vortex lines are structurally achiral regions in which a molecular twist cannot be defined. We can reversibly switch between fusion and fission of these vortex knots by applying electric pulses. This reveals the physical embodiments of concepts in knot theory, such as connected sums of knots and band surgeries. Our findings demonstrate the interplay of chirality effects at hierarchical levels from constituent molecules to the host medium and the energetically stable chiral vortex knots. This emergent physical behaviour may enable applications in electro-optics and photonics in which such fusion and fission processes of vortex knots can be used for controlling light.

Applied mathematics↗

DOE OSTI2021

Symbolic pregression: Discovering physical laws from distorted video

In this work, we present a method for unsupervised learning of equations of motion for objects in raw and optionally distorted unlabeled synthetic video (or, more generally, for discovering and modeling predictable features in time-series data). We first train an autoencoder that maps each video frame into a low-dimensional latent space where the laws of motion are as simple as possible, by minimizing a combination of nonlinearity, acceleration, and prediction error. Differential equations describing the motion are then discovered using Pareto-optimal symbolic regression. We find that our pre-regression (“pregression”) step is able to rediscover Cartesian coordinates of unlabeled moving objects even when the video is distorted by a generalized lens. Using intuition from multidimensional knot theory, we find that the pregression step is facilitated by first adding extra latent space dimensions to avoid topological problems during training and then removing these extra dimensions via principal component analysis. An inertial frame is autodiscovered by minimizing the combined equation complexity for multiple experiments.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2022

Topological Link Models of Multipartite Entanglement

We introduce a novel model of multipartite entanglement based on topological links, generalizing the graph/hypergraph entropy cone program. We demonstrate that there exist link representations of entropy vectors which provably cannot be represented by graphs or hypergraphs. Furthermore, we show that the contraction map proof method generalizes to the topological setting, though now requiring oracular solutions to well-known but difficult problems in knot theory.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2025

Decay of skeins of dislocations in cholesterics: rewiring Conway's tangles into necklaces of bangles

Knotted and linked skeins of vortices and disclinations generated, respectively, by symmetry-breaking normal → superfluid and isotropic → nematic phase transitions are known to untie, by rewiring of their crossings, into independent unknots that finally shrink and collapse until the defect-free ground state is reached. We demonstrate that the decay of skeins of dislocations, generated by the isotropic → cholesteric phase transition within a cylinder/cylinder gap, leads to stable necklace-like states made of numerous minimal loops, called bangles, tethered to kinks of much larger loops called cargo. We analyze the topological decay of skeins of dislocations in terms of the Conway–Kauffman theory of knots, showing that the necklace state results from rewiring of crossings triggered by collisions of tangles with their numerator closure. We point out that, in general, for symmetry reasons, kinks on edge dislocations are chiral. Their handedness, right or left, directly depends on the sign of kinks on which they are localized. In cholesterics with intrinsic chirality, the energy of kinks bearing bangles depends on their handedness. For this reason, within necklaces, all bangles are tethered to kinks of the same sign.

Lee, Jun-Yong [International Institute for Sustain↗

DOE OSTI2020

Review: knots and other new topological effects in liquid crystals and colloids

Abstract Humankind has been obsessed with knots in religion, culture and daily life for millennia, while physicists like Gauss, Kelvin and Maxwell already involved them in models centuries ago. Nowadays, colloidal particles can be fabricated to have shapes of knots and links with arbitrary complexity. In liquid crystals, closed loops of singular vortex lines can be knotted by using colloidal particles and laser tweezers, as well as by confining nematic fluids into micrometer-sized droplets with complex topology. Knotted and linked colloidal particles induce knots and links of singular defects, which can be interlinked (or not) with colloidal particle knots, revealing the diversity of interactions between topologies of knotted fields and topologically nontrivial surfaces of colloidal objects. Even more diverse knotted structures emerge in nonsingular molecular alignment and magnetization fields in liquid crystals and colloidal ferromagnets. The topological solitons include hopfions, skyrmions, heliknotons, torons and other spatially localized continuous structures, which are classified based on homotopy theory, characterized by integer-valued topological invariants and often contain knotted or linked preimages, nonsingular regions of space corresponding to single points of the order parameter space. A zoo of topological solitons in liquid crystals, colloids and ferromagnets promises new breeds of information displays and a plethora of data storage, electro-optic and photonic applications. Their particle-like collective dynamics echoes coherent motions in active matter, ranging from crowds of people to schools of fish. This review discusses the state of the art in the field, as well as highlights recent developments and open questions in physics of knotted soft matter. We systematically overview knotted field configurations, the allowed transformations between them, their physical stability and how one can use one form of knotted fields to model, create and imprint other forms. The large variety of symmetries accessible to liquid crystals and colloids offer insights into stability, transformation and emergent dynamics of fully nonsingular and singular knotted fields of fundamental and applied importance. The common thread of this review is the ability to experimentally visualize these knots in real space. The review concludes with a discussion of how the studies of knots in liquid crystals and colloids can offer insights into topologically related structures in other branches of physics, with answers to many open questions, as well as how these experimentally observable knots hold a strong potential for providing new inspirations to the mathematical knot theory.

Physics↗

DOE OSTI2021

Rozansky-Witten geometry of Coulomb branches and logarithmic knot invariants

By studying Rozansky-Witten theory with non-compact target spaces we find new connections with knot invariants whose physical interpretation was not known. Furthermore, this opens up several new avenues, which include a new formulation of q-series invariants of 3-manifolds in terms of affine Grassmannians and a generalization of Akutsu-Deguchi-Ohtsuki knot invariants.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1980

Full scale wind tunnel investigation of a bearingless main helicopter rotor

A stability test program was conducted to determine the effects of airspeed, collective pitch, rotor speed and shaft angle on stability and loads at speeds beyond that attained in the BMR/BO-105 flight test program. Loads and performance data were gathered at forward speeds up to 165 knots. The effect of cyclic pitch perturbations on rotor response was investigated at simulated level flight conditions. Two configuration variations were tested for their effect on stability. One variable was the control system stiffness. An axially softer pitch link was installed in place of the standard BO-105 pitch link. The second variation was the addition of elastomeric damper strips to increase the structural damping. The BMR was stable at all conditions tested. At fixed collective pitch, shaft angle and rotor speed, damping generally increased between hover and 60 knots, remained relatively constant from 60 to 90 knots, then decreased above 90 knots. Analytical predictions are in good agreement with test data up to 90 knots, but the trend of decreasing damping above 90 knots is contrary to the theory.

Source record↗

NASA NTRS1978

EUV structure of a small flare

Observations of a small flare are presented using data from the Harvard spectroheliometer on Skylab. The event is discussed in terms of the magnetic structure of the active region as deduced from the EUV observations and from field line extrapolations. The role of emerging flux in the initial flare brightenings is emphasized. A detailed model of one loop is deduced using the EUV data. This self-consistent model indicates initial heating of the loop modeled near its top, and mass flow into the cool core of the loop, with matter preferentially concentrating in a few distinct knots along the loop. Implications for theories of the flare process are discussed.

Levine, R. H.↗

DOE OSTI2023

Non-Semisimple TQFT's and BPS q -Series

We propose and in some cases prove a precise relation between 3-manifold invariants associated with quantum groups at roots of unity and at generic q. Both types of invariants are labeled by extra data which plays an important role in the proposed relation. Bridging the two sides - which until recently were developed independently, using very different methods - opens many new avenues. In one direction, it allows to study (and perhaps even to formulate) q-series invariants labeled by spin c structures in terms of non-semisimple invariants. In the opposite direction, it offers new insights and perspectives on various elements of non-semisimple TQFT's, bringing the latter into one unifying framework with other invariants of knots and 3-manifolds that recently found realization in quantum field theory and in string theory.

3-manifold invariants↗

DOE OSTI2022

Random Knotting in Fractal Ring Polymers

Many ring polymer systems of physical and biological interest exhibit both pronounced topological effects and nontrivial self-similarity, but the relationship between these two phenomena has not yet been clearly established. Here, we use theory and simulation to formulate such a connection by studying a fundamental topological property—the random knotting probability—for ring polymers with varying fractal dimension, d f . Using straightforward scaling arguments, we generalize a classic mathematical result, showing that the probability of a trivial knot decays exponentially with chain size, N, for all fractal dimensions: P 0 (N) ∝ exp(–N/N 0 ). However, no such simple considerations can account for the dependence of the knotting length, N 0 , on d f , necessitating a more involved analytical calculation. This analysis reveals a complicated double-exponential dependence, which is well supported by numerical data. By contrast, functional forms typical of simple scaling theories fail to adequately describe the observations. These findings are equally valid for two-dimensional ring polymer systems, where “knotting” is defined as the intersection of any two segments.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

NASA NTRS1981

RSRA vertical drag test report

The Rotor Systems Research Aircraft (RSRA), because of its ability to measure rotor loads, was used to conduct an experiment to determine vertical drag, tail rotor blockage, and thrust augmentation as affected by ground clearance and flight velocity. The RSRA was flown in the helicopter configuration at speeds from 0 to 15 knots for wheel heights from 5 to 150 feet, and to 60 knots out of ground effect. The vertical drag trends in hover, predicted by theory and shown in model tests, were generally confirmed. The OGE hover vertical drag is 4.0 percent, 1.1 percent greater than predicted. The vertical drag decreases rapidly as wheel height is reduced, and is zero at a wheel height of 6 feet. The vertical drag also decreases with forward speed, approaching zero at sixty knots. The test data show the effect of wheel height and forward speed on thrust, gross weight capability, and power, and provide the relationships for power and collective pitch at constant gross weight required for the simulation of helicopter takeoffs and landings.

Flemming, R. J.↗

NASA NTRS1983

Velocity dispersions of knots in Vela X and Puppis A

Very high resolution spectroscopy of optical emission lines indicates that the velocity dispersions of knots in Vela X and Puppis A result primarily from turbulence within the emitting regions. Line-of-sight velocity dispersions (half-widths at half-maximum) for the observed knots in both remnants are about 20-30 km/s. In Vela X, these include thermal velocity dispersions of roughly 16 km/s for hydrogen and 4 km/s for nitrogen. The knots themselves move randomly relative to each other, with speeds up to 30 km/s in Vela X and between 30-80 km/s in Puppis A. Occasionally, diffuse components of the filaments are observed in Puppis A. These diffuse components have velocity dispersions between 55 and 80 km/s, but the relative contributions of turbulent and thermal motions to the dispersions are unknown. The results are compared with theory and previous observations. Direct photographs may be able to show if the forbidden O III knots are less spatially dispersed than the forbidden N II, forbidden S II, forbidden O II, and H I knots.

Shull, P., Jr.↗

DOE OSTI2024

3d gravity from Virasoro TQFT: Holography, wormholes and knots

We further develop the description of three-dimensional quantum gravity with negative cosmological constant in terms of Virasoro TQFT formulated in our previous paper [SciPost Phys. 15, 151 (2023)]. We compare the partition functions computed in the Virasoro TQFT formalism to the semiclassical evaluation of Euclidean gravity partition functions. This matching is highly non-trivial, but can be checked directly in some examples. We then showcase the formalism in action, by computing the gravity partition functions of many relevant topologies. For holographic applications, we focus on the partition functions of Euclidean multi-boundary wormholes with three-punctured spheres as boundaries. This precisely quantifies the higher moments of the structure constants in the proposed ensemble boundary dual and subjects the proposal to thorough checks. Finally, we investigate in detail the example of the figure eight knot complement as a hyperbolic 3-manifold. We show that the Virasoro TQFT partition function is identical to the partition function computed in Teichmüller theory, thus giving strong evidence for the equivalence of these TQFTs. We also show how to produce a large class of manifolds via Dehn surgery on the figure eight knot.

Collier, Scott (ORCID:0000000286476653)↗

DOE OSTI2024

What are the Principles Controlling Biomimetic Heteropolymer Secondary Structure? (Final Technical Report)

The goal of the project was to develop improved theories to understand how nonbiological oligomers could be designed to cooperatively fold into 3D structures. These studies would lay the groundwork for materials made of such molecules, making it possible to create controlled and ordered materials for electron transport, efficient protein-like catalysts that work under extreme conditions, and sensors with highly-specific chemical responsiveness. Two different simulation thrusts were investigated, one focused on programs to identify stable low energy folded structures at a coarse-level of description of oligomers, and another to calculate thermodynamics of such oligomers. We used these theories to answer several specific questions about what properties of oligomers lead to cooperative transitions, and to identify how oligomer knots could serve as secondary structure elements. We also carried out significant collaborative investigation with Dr. Samuel Gellman (UW-Madison, National Academy of Sciences member) on stability for foldamers of interest to them. Only one of the experimentally tested foldamers stably folded, which was indicated by simulations as being the most likely to fold. Finally, we developed new theoretical descriptions of foldamers, showing how cooperativity was determined primarily by the entropy difference between the folded and unfolded state. The research did not answer all questions laid out in the original proposal but laid the groundwork for later efforts to design folded oligomers materials with high switchability.

36 MATERIALS SCIENCE↗

NASA NTRS1990

Acoustic results of the Boeing model 360 whirl tower test

An evaluation is presented for whirl tower test results of the Model 360 helicopter's advanced, high-performance four-bladed composite rotor system intended to facilitate over-200-knot flight. During these performance measurements, acoustic data were acquired by seven microphones. A comparison of whirl-tower tests with theory indicate that theoretical prediction accuracies vary with both microphone position and the inclusion of ground reflection. Prediction errors varied from 0 to 40 percent of the measured signal-to-peak amplitude.

Watts, Michael E.↗

NASA NTRS1993

Models of the Cartwheel ring galaxy: Spokes and starbursts

Recent observations of this famous ring galaxy, including optical and near-infrared CCD surface photometry, and VLA radio continuum and 21 cm line mapping (Higdon 1992b, in prep.), have inspired a renewed modeling effort. Toomre's (1978, in The Large-scale Structure of the Universe, eds. Longair and Einasto) series of restricted three-body simulations demonstrated how the multiple rings could be produced in a nearly head-on galaxy collision. New models with a halo-dominated potential based on the 21 cm rotation curve are able to reproduce such details as the spacing between rings, ring widths, offset of the nucleus, and several kinematical features, thus providing strong support for the collisional theory. The new observations have shown there are little or no old stars in Cartwheel; it may consist almost entirely of gas and stars produced as a result of compression in the ring wave. To model this process Smooth Particle Hydrodynamics (SPH) simulations of the Cartwheel disk have been performed. Fixed gravitational potentials were used to represent the Cartwheel and a roughly 30 percent mass collision partner. The interaction dynamics was treated as in the usual restricted three-body approximation, and the effects of local self-gravity between disk particles were calculated. We are particularly interested in testing the theory that enhanced star formation in waves is the result of gravitational instability in the compressed region (see e.g. Kennicutt 1989, ApJ 344, 685). The gas surface density in a number of simulations was initialized to a value slightly below the threshold for local gravitational instability throughout most of the disk. The first ring wave produces relatively modest compressions (a factor of order a few), triggering instability in a narrow range of wavelengths. Self-gravity in the disk is calculated over a comparable range of scales. Simulations were run with isothermal, adiabatic, and adiabatic with radiative cooling characterized by a relatively short timescale. The isothermal approximation is good except in the vicinity of the strong second (inner) ring, and several snapshots from one case are shown in the figure below. Flocculent spiral segments are present before the collision, and these are compressed into dense knots in the ring wave. These knots are likely to be sites of vigorous star formation. In the strong rarefaction behind the outer ring most of the knots are radially stretched and sheared, giving rise to spoke-like features. A few dense knots are evidently very tightly bound, because they retain their coherence and are stretched relatively little through the rarefaction. This is in accord with evidence for continuing star formation in some spokes (Marcum, Appleton and Higdon 1992). The number and spacing of spokes is a direct function of the scale of the gravitational instability in the disk. Thus, the gravitational instability theory, together with the hypothesis that massive stars are only formed in dense knots of gas, can account for most of the distinct morphology of the Cartwheel.

Struck-Marcell, Curtis↗