Hopf Link Research Articles

Knotted trajectories of neutral and charged particles in Gaussian light beams

Making use of the equivalence between paraxial wave equation and two-dimensional Schr\"odinger equation, Gaussian beams of monochromatic light, possessing knotted nodal structures are obtained in an analytical way. These beams belong to the wide class of paraxial beams called the Hypergeometric-Gaussian beams [E. Karimi, G. Zito, B. Piccirillo, L. Marrucci and E. Santamato, Opt. Lett. {\bf 32}, 3053(2007)]. Four topologies are dealt with: the unknot, the Hopf link, the Borromean rings and the trefoil. It is shown in the numerical way that neutral polarizable particles placed in such light fields, upon precise tuning of the initial conditions, can be forced to follow the identical knotted trajectories. A similar outcome is also valid for charged particles that are subject to a ponderomotive potential. This effect can serve to precisely steer particles along chosen complicated pathways exhibiting non-trivial topological character, guide them around obstacles and seems to be helpful in engineering more complex nanoparticles.

Are Maxwell knots integrable?

We review properties of the null-field solutions of source-free Maxwell equations. We focus on the electric and magnetic field lines, especially on limit cycles, which actually can be knotted and/or linked at every given moment. We analyse the fact that the Poynting vector induces self-consistent time evolution of these lines and demonstrate that the Abelian link invariant is integral of motion. We also consider particular examples of the field lines for the particular family of finite energy source-free “knot” solutions, attempting to understand when the field lines are closed – and can be discussed in terms of knots and links. Based on computer simulations we conjecture that Ranada’s solution, where every pair of lines forms a Hopf link, is rather exceptional. In general, only particular lines (a set of measure zero) are limit cycles and represent closed lines forming knots/links, while all the rest are twisting around them and remain unclosed. Still, conservation laws of Poynting evolution and associated integrable structure should persist.

Legendrian Hopf Links

Abstract We completely classify Legendrian realizations of the Hopf link, up to coarse equivalence, in the 3-sphere with any contact structure.

Irreversible Dynamics of Vortex Reconnections in Quantum Fluids.

We statistically study vortex reconnections in quantum fluids by evolving different realizations of vortex Hopf links using the Gross-Pitaevskii model. Despite the time reversibility of the model, we report clear evidence that the dynamics of the reconnection process is time irreversible, as reconnecting vortices tend to separate faster than they approach. Thanks to a matching theory devised concurrently by Proment and Krstulovic [Phys. Rev. Fluids 5, 104701 (2020)PLFHBR2469-990X10.1103/PhysRevFluids.5.104701], we quantitatively relate the origin of this asymmetry to the generation of a sound pulse after the reconnection event. Our results have the prospect of being tested in several quantum fluid experiments and, theoretically, may shed new light on the energy transfer mechanisms in both classical and quantum turbulent fluids.

Hopf superpolynomial from topological vertices

Link/knot invariants are series with integer coefficients, and it is a long-standing problem to get them positive and possessing cohomological interpretation. Constructing positive “superpolynomials” is not straightforward, especially for colored invariants. A simpler alternative is a multi-parametric generalization of the character expansion, which leads to colored “hyperpolynomials”. The third construction involves branes on resolved conifolds, which gives rise to still another family of invariants associated with composite representations. We revisit this triality issue in the simple case of the Hopf link and discover a previously overlooked way to produce positive colored superpolynomials from the DIM-governed four-point functions, thus paving a way to a new relation between super- and hyperpolynomials.

Hopf link volume simplicity constraints in spin foam models

In this article we consider specific bivector geometries which arise in the large-spin limit of the extension of the Engle–Pereira–Rovelli–Livine spin foam model for quantum gravity by Kaminski, Kisielowski and Lewandowski. We address the implementation of volume simplicity constraints, which are required to ensure that a 4d metric can be reconstructed from the bivector geometry. We find that the necessary conditions are closely related, but not quite equal to the Hopf link volume simplicity constraints introduced in earlier works. We estimate the number of independent geometricity conditions for arbitrary bivector geometries, and find that they always agree with the number of Hopf links on the graph minus one, suggesting that the geometricity conditions can generically be formulated by deformation of the Hopf link volume simplicity constraints.

Geodesic fibrations for packing diabolic domains

We describe a theory of packing hyperboloid "diabolic" domains in bend-free textures of liquid crystals. The domains sew together continuously, providing a menagerie of bend-free textures akin to the packing of focal conic domains in smectic liquid crystals. We show how distinct domains may be related to each other by Lorentz transformations and that this process may lower the elastic energy of the system. We discuss a number of phases that may be formed as a result, including splay-twist analogues of blue phases. We also discuss how these diabolic domains may be subject to "superluminal boosts," yielding defects analogous to shock waves. We explore the geometry of these textures, demonstrating their relation to Milnor fibrations of the Hopf link. Finally, we show how the theory of these domains is unified in four-dimensional space.

Visualizing one-dimensional non-hermitian topological phases

We develop a graphic approach for characterizing one-dimensional non-Hermitian topological phases. The eigenstates of energy bands are mapped to a graph on the torus, where a nontrivial topology exhibits as links. The topology of band touching exceptional points is a crucial aspect of a non-Hermitian system; the existence of exceptional point results in networks. We discuss the parity-time () symmetric two-band models. The pseudo-anti-Hermiticity protects the band topology, and the eigenstate graphs in the exact -symmetric phase locate on the torus surface under the symmetry protection. For the Su-Schrieffer-Heeger ladder, the eigenstate graph is a Hopf link in the gapped nontrivial phase; chiral-time symmetry protects that the movable exceptional points appear in pairs in the real-energy gapless phase, and each exceptional point splits into a pair of exceptional points when the symmetry breaks. The proposed graphic approach is applicable in one-dimensional N-band models. Our findings provide insight into one-dimensional non-Hermitian topology phases through visualizing the eigenstates.

Creation of acoustic vortex knots

Knots and links have been conjectured to play a fundamental role in a wide range of scientific fields. Recently, tying isolated vortex knots in the complex optical field has been realized. However, how to construct the acoustic vortex knot is still an unknown problem. Here we propose theoretically and demonstrate experimentally the creation of acoustic vortex knots using metamaterials, with decoupled modulation of transmitted phase and amplitude. Based on the numerical simulation, we find that the knot function can be embedded into the acoustic field by designed metamaterials with only 24 × 24 pixels. Furthermore, using the optimized metamaterials, the acoustic fields with Hopf link and trefoil knot vortex lines have been observed experimentally.

Double exceptional links in a three-dimensional dissipative cold atomic gas

We explore the topological properties of non-Hermitian nodal-link semimetals with dissipative cold atoms in a three-dimensional optical lattice. We construct a two-band continuum model in three dimensions with a spin-dependent gain and loss, where the exceptional points in the energy spectrum can comprise a double Hopf link. The topology of the bulk band is characterized by a winding number defined for a one-dimensional loop in the momentum space and a topological transition of the nodal structures emerges as the change of the non-Hermiticity strength. A non-Bloch theory is built to describe the corresponding lattice model which has anomalous bulk-boundary correspondence. Furthermore, we propose that the model can be realized using ultracold fermionic atoms in an optical lattice and the exceptional nodal links as well as the topological properties can be detected by measuring the atomic spin textures.

Matrix models for classical groups and Toeplitz ± Hankel minors with applications to Chern–Simons theory and fermionic models

We study matrix integration over the classical Lie groups U(N), Sp(2N), SO(2N) and SO(2N + 1), using symmetric function theory and the equivalent formulation in terms of determinants and minors of Toeplitz ± Hankel matrices. We establish a number of factorizations and expansions for such integrals, also with insertions of irreducible characters. As a specific example, we compute both at finite and large N the partition functions, Wilson loops and Hopf links of Chern–Simons theory on S3 with the aforementioned symmetry groups. The identities found for the general models translate in this context to relations between observables of the theory. Finally, we use character expansions to evaluate averages in random matrix ensembles of Chern–Simons type, describing the spectra of solvable fermionic models with matrix degrees of freedom.

On Cayley graphs of {\bb Z}^4.

The generating sets of {\bb Z}^4 have been enumerated which consist of integral four-dimensional vectors with components -1, 0, 1 and allow Cayley graphs without edge intersections in a straight-edge embedding in a four-dimensional Euclidean space. Owing to computational restrictions the valency of enumerated graphs has been fixed to 10. Up to isomorphism 58 graphs have been found and characterized by coordination sequences, shortest cycles and automorphism groups. To compute automorphism groups, a novel strategy is introduced that is based on determining vertex stabilizers from the automorphism group of a sufficiently large finite ball cut out from an infinite graph. Six exceptional, rather `dense' graphs have been identified which are locally isomorphic to a five-dimensional cubic lattice within a ball of radius 10. They could be built by either interconnecting interpenetrated three- or four-dimensional cubic lattices and therefore necessarily contain Hopf links between quadrangular cycles. As a consequence, a local combinatorial isomorphism does not extend to a local isotopy.

Jones Polynomial and Knot Transitions in Hermitian and non-Hermitian Topological Semimetals.

Topological nodal line semimetals host stable chained, linked, or knotted line degeneracies in momentum space protected by symmetries. In this Letter, we use the Jones polynomial as a general topological invariant to capture the global knot topology of the oriented nodal lines. We show that every possible change in Jones polynomial is attributed to the local evolutions around every point where two nodal lines touch. As an application of our theory, we show that nodal chain semimetals with four touching points can evolve to a Hopf link. We extend our theory to 3D non-Hermitian multiband exceptional line semimetals. Our work provides a recipe to understand the transition of the knot topology for protected nodal lines.

Topological carbon allotropes: knotted molecules, carbon-nano-chain, chainmails, and Hopfene

Carbon allotropes such as diamond, nano-tube, Fullerene, and Graphene were discovered and revolutionised material sciences. These structures have unique translational and rotational symmetries, described by a crystallographic group theory, and the atoms are arranged at specific rigid positions in 3-dimensional (D) space. Regardless of these exotic molecular structures, the structures of materials are topologically trivial in a mathematical sense, that their bonds are connected without a link nor a knot. These days, the progress on the synthetic chemistry is significant to make various topologically non-trivial molecular structures. Topological molecules (0D) including Trefoil knots, a Hopf-link, a Möbius strip, and Borromean rings, were already realised. However, their potentially exotic electronic properties have not been sufficiently explored. Here, we propose a new 3D carbon allotrope, named Hopfene, which has periodic arrays of Hopf-links to knit horizontal Graphene sheets into vertical ones without connecting by σ bonds. We conducted an ab inito band structure calculation using a Density-Functional-Theory (DFT) for Hopfene, and found that it is well-described by a tight-binding model. We confirmed the original Dirac points of 2D Graphene were topologically protected upon the introduction of the Hopf links, and low-energy excitations are described by 1D, 2D, and 3D gapless Fermions.

The First Quantitative Synthesis of a Closed Three-Link Chain (613) Using Coordination and Noncovalent Interactions-Driven Self-Assembly.

Engineering of supramolecular topologies offers potential opportunities for tailoring their properties to various function and applications. However, the synthesis of interlocked or intertwined compounds, catenanes, links or knots, is a challenge. Previously, we used coordination-driven self-assembly and noncovalent interactions (NCIs) between metal-based acceptors and multipyridyl donors to create supramolecular topologies with increasing complexity. Self-assembling components of fixed length and geometry have been utilized for the production of topologies such as Borromean rings, Solomon links, Hopf's link, "rectangle in rectangle", and an 818 molecular knot. However, recent synthesis of a linear [3]catenane by us witnessed the importance of flexible ligands along with coordination-driven self-assembly and NCIs in self-assembling units. This flexibility provides distinctive angularity for the recognition of various NCIs and thus offers tremendous possibilities for realizing complex supramolecular topologies. This study proposed a selective and quantitative synthesis, and also the first X-ray characterization of a closed three-link chain (a prime link of [3]catenane with 6 crossings) via two component coordination-driven self-assembly. The experiments based upon concentration, guest template, and solvent effects were systematically presented. Furthermore, the experimental finding was supported by density functional theory calculations, which highlighted the necessity of the multiple NCIs along with appropriate geometry of the [2 + 2] rings.

Braided Tensor Categories Related to $${\mathcal {B}}_{p}$$ Vertex Algebras

The $${\mathcal {B}}_{p}$$ -algebras are a family of vertex operator algebras parameterized by $$p\in {\mathbb {Z}}_{\ge 2}$$ . They are important examples of logarithmic CFTs and appear as chiral algebras of type $$(A_1, A_{2p-3})$$ Argyres–Douglas theories. The first member of this series, the $${\mathcal {B}}_2$$ -algebra, are the well-known symplectic bosons also often called the $$\beta \gamma $$ vertex operator algebra. We study categories related to the $${\mathcal {B}}_{p}$$ vertex operator algebras using their conjectural relation to unrolled restricted quantum groups of $${\mathfrak {sl}}_{2}$$ . These categories are braided, rigid and non semi-simple tensor categories. We list their simple and projective objects, their tensor products and their Hopf links. The latter are succesfully compared to modular data of characters thus confirming a proposed Verlinde formula of David Ridout and the second author.

Projective objects and the modified trace in factorisable finite tensor categories

For ${\mathcal{C}}$ a factorisable and pivotal finite tensor category over an algebraically closed field of characteristic zero we show:(1)${\mathcal{C}}$ always contains a simple projective object;(2)if ${\mathcal{C}}$ is in addition ribbon, the internal characters of projective modules span a submodule for the projective $\text{SL}(2,\mathbb{Z})$-action;(3)the action of the Grothendieck ring of ${\mathcal{C}}$ on the span of internal characters of projective objects can be diagonalised;(4)the linearised Grothendieck ring of ${\mathcal{C}}$ is semisimple if and only if ${\mathcal{C}}$ is semisimple.Results (1)–(3) remain true in positive characteristic under an extra assumption. Result (1) implies that the tensor ideal of projective objects in ${\mathcal{C}}$ carries a unique-up-to-scalars modified trace function. We express the modified trace of open Hopf links coloured by projectives in terms of $S$-matrix elements. Furthermore, we give a Verlinde-like formula for the decomposition of tensor products of projective objects which uses only the modular $S$-transformation restricted to internal characters of projective objects. We compute the modified trace in the example of symplectic fermion categories, and we illustrate how the Verlinde-like formula for projective objects can be applied there.

A note on the S-dual basis in the free fermion system

Abstract The free fermion system is the simplest quantum field theory which has the symmetry of the Ding–Iohara–Miki algebra (DIM). DIM has S-duality symmetry, known as Miki automorphism, which defines the transformation of generators. We introduce the second set of the fermionic basis (S-dual basis) which implements the duality transformation. It may be interpreted as the Fourier dual of the standard basis, and the inner product between the standard and the S-dual is proportional to the Hopf link invariant. We also rewrite the general topological vertex in the form of an Awata–Feigin–Shiraishi intertwiner and show that it becomes more symmetric for the duality transformation.

Triple-crossing number and moves on triple-crossing link diagrams

Every link in the 3-sphere has a projection to the plane where the only singularities are pairwise transverse triple points. The associated diagram, with height information at each triple point, is a triple-crossing diagram of the link. We give a set of diagrammatic moves on triple-crossing diagrams analogous to the Reidemeister moves on ordinary diagrams. The existence of [Formula: see text]-crossing diagrams for every [Formula: see text] greater than one allows the definition of the [Formula: see text]-crossing number. We prove that for any nontrivial, nonsplit link, other than the Hopf link, its triple-crossing number is strictly greater than its quintuple-crossing number.

Generalizations of the Conway–Gordon theorems and intrinsic knotting on complete graphs

In 1983, Conway and Gordon proved that for every spatial complete graph on six vertices, the sum of the linking numbers over all of the constituent two-component links is odd, and that for every spatial complete graph on seven vertices, the sum of the Arf invariants over all of the Hamiltonian knots is odd. In 2009, the second author gave integral lifts of the Conway–Gordon theorems in terms of the square of the linking number and the second coefficient of the Conway polynomial. In this paper, we generalize the integral Conway–Gordon theorems to complete graphs with arbitrary number of vertices greater than or equal to six. As an application, we show that for every rectilinear spatial complete graph whose number of vertices is greater than or equal to six, the sum of the second coefficients of the Conway polynomials over all of the Hamiltonian knots is determined explicitly in terms of the number of triangle-triangle Hopf links.