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Results for “knot theory”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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17 records
All sourcesNASA NTRSDOE 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↗
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↗
DOE OSTI2020

Refined large N duality for knots

We formulate large [Formula: see text] duality of [Formula: see text] refined Chern–Simons theory with a torus knot/link in [Formula: see text]. By studying refined BPS states in M-theory, we provide the explicit form of low-energy effective actions of Type IIA string theory with D4-branes on the [Formula: see text]-background. This form enables us to relate refined Chern–Simons invariants of a torus knot/link in [Formula: see text] to refined BPS invariants in the resolved conifold. Assuming that the extra [Formula: see text] global symmetry acts on BPS states trivially, the duality predicts graded dimensions of cohomology groups of moduli spaces of M2–M5 bound states associated to a torus knot/link in the resolved conifold. Thus, this formulation can be also interpreted as a positivity conjecture of refined Chern–Simons invariants of torus knots/links. We also discuss about an extension to non-torus knots.

Mathematics↗
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↗
DOE OSTI2022

Branches, quivers, and ideals for knot complements

We generalize the F K invariant, i.e. Z^ for the complement of a knot K in the 3-sphere, the knots-quivers correspondence, and A-polynomials of knots, and find several interconnections between them. We associate an F K invariant to any branch of the A-polynomial of K and we work out explicit expressions for several simple knots. We show that these F K invariants can be written in the form of a quiver generating series, in analogy with the knots-quivers correspondence. We discuss various methods to obtain such quiver representations, among others using R-matrices. We generalize the quantum a-deformed A-polynomial to an ideal that contains the recursion relation in the group rank, i.e. in the parameter a, and describe its classical limit in terms of the Coulomb branch of a 3d-5d theory. We also provide t-deformed versions. Furthermore, we study how the quiver formulation for closed 3-manifolds obtained by surgery leads to the superpotential of 3d N = 2 theory T[M 3 ] and to the data of the associated modular tensor category MTC[M 3 ].

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

3d-3d correspondence for mapping tori

One of the main challenges in 3d-3d correspondence is that no existent approach offers a complete description of 3d N = 2 SCFT T [ M 3 ] — or, rather, a “collection of SCFTs” as we refer to it in the paper — for all types of 3-manifolds that include, for example, a 3-torus, Brieskorn spheres, and hyperbolic surgeries on knots. The goal of this paper is to overcome this challenge by a more systematic study of 3d-3d correspondence that, first of all, does not rely heavily on any geometric structure on M 3 and, secondly, is not limited to a particular supersymmetric partition function of T [ M 3 ]. In particular, we propose to describe such “collection of SCFTs” in terms of 3d N = 2 gauge theories with “non-linear matter” fields valued in complex group manifolds. As a result, we are able to recover familiar 3-manifold invariants, such as Turaev torsion and WRT invariants, from twisted indices and half-indices of T [ M 3 ], and propose new tools to compute more recent q -series invariants Z * ( M 3 ) in the case of manifolds with b 1 > 0. Although we use genus-1 mapping tori as our “case study,” many results and techniques readily apply to more general 3-manifolds, as we illustrate throughout the paper.

Topological Field Theories↗
DOE OSTI2022

Hopfions, heliknotons, skyrmions, torons and both abelian and nonabelian vortices in chiral liquid crystals

Widely known for their uses in displays and electro-optics, liquid crystals are more than just technological marvels. They vividly reveal the topology and structure of various solitonic and singular field configurations, often markedly resembling the ones arising in many field theories and in the areas ranging from particle physics to optics, hard condensed matter and cosmology. In this review, we focus on chiral nematic liquid crystals to show how these experimentally highly accessible systems provide valuable insights into the structure and behavior of fractional, full, and multi-integer two-dimensional skyrmions, dislocations and both abelian and non-abelian defect lines, as well as various three-dimensionally localized, often knotted structures that include hopfions, heliknotons, torons and twistions. Finally, we provide comparisons of some of these field configurations with their topological counterparts in chiral magnets, discussing close analogies between these two condensed matter systems.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

The galaxy–halo connection of emission-line galaxies in IllustrisTNG

ABSTRACT We employ the hydrodynamical simulation IllustrisTNG-300-1 to explore the halo occupation distribution (HOD) and environmental dependence of luminous star-forming emission-line galaxies (ELGs) at z ∼ 1. Such galaxies are key targets for current and upcoming cosmological surveys. We select model galaxies through cuts in colour–colour space allowing for a direct comparison with the Extended Baryon Oscillation Spectroscopic Survey and the Dark Energy Spectroscopic Instrument (DESI) surveys and then compare them with galaxies selected based on specific star formation rate (sSFR) and stellar mass. We demonstrate that the ELG populations are twice more likely to reside in lower density regions (sheets) compared with the mass-selected populations and twice less likely to occupy the densest regions of the cosmic web (knots). We also show that the colour-selected and sSFR-selected ELGs exhibit very similar occupation and clustering statistics, finding that the agreement is best for lower redshifts. In contrast with the mass-selected sample, the occupation of haloes by a central ELG peaks at ∼20 per cent. We furthermore explore the dependence of the HOD and the autocorrelation on environment, noticing that at fixed halo mass, galaxies in high-density regions cluster about 10 times more strongly than low-density ones. This result suggests that we should model carefully the galaxy–halo relation and implement assembly bias effects into our models (estimated at ∼4 per cent of the clustering of the DESI colour-selected sample at z = 0.8). Finally, we apply a simple mock recipe to recover the clustering on large scales (r ≳ 1 Mpc h−1) to within 1 per cent by augmenting the HOD model with an environment dependence, demonstrating the power of adopting flexible population models.

79 ASTRONOMY AND ASTROPHYSICS↗