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Effects connected with the mathematical theory of knots 1 emerge in many areas of science, from physics 2 , 3 to biology 4 . Recent theoretical work discovered that the braid group characterizes the topology of non-Hermitian periodic systems 5 , where the complex band energies can braid in momentum space. However, such braids of complex-energy bands have not been realized or controlled experimentally. Here, we introduce a tight-binding lattice model that can achieve arbitrary elements in the braid group of two strands 𝔹 2 . We experimentally demonstrate such topological complex-energy braiding of non-Hermitian bands in a synthetic dimension 6 , 7 . Our experiments utilize frequency modes in two coupled ring resonators, one of which undergoes simultaneous phase and amplitude modulation. We observe a wide variety of two-band braiding structures that constitute representative instances of links and knots, including the unlink, the unknot, the Hopf link and the trefoil. We also show that the handedness of braids can be changed. Our results provide a direct demonstration of the braid-group characterization of non-Hermitian topology and open a pathway for designing and realizing topologically robust phases in open classical and quantum systems. Experiments using two coupled optical ring resonators and based on the concept of synthetic dimension reveal non-Hermitian energy band structures exhibiting topologically non-trivial knots and links.

Although free groups have long been present as part of algebra, and it steady has its implications for each implementation target, especially to determine invariant. However, the decrease in free groups of webbing requires different meanings of braid. This paper reveals a concept to form a free group of braid groups.

A graph is Helly if every family of pairwise intersecting combinatorial balls has a nonempty intersection. We show that weak Garside groups of finite type and FC-type Artin groups are Helly, that is, they act geometrically on Helly graphs. In particular, such groups act geometrically on spaces with a convex geodesic bicombing, equipping them with a nonpositive-curvature-like structure. That structure has many properties of a CAT(0) structure and, additionally, it has a combinatorial flavor implying biautomaticity. As immediate consequences we obtain new results for FC-type Artin groups (in particular braid groups and spherical Artin groups) and weak Garside groups, including e.g. fundamental groups of the complements of complexified finite simplicial arrangements of hyperplanes, braid groups of well-generated complex reflection groups, and one-relator groups with non-trivial center. Among the results are: biautomaticity, existence of EZ and Tits boundaries, the Farrell–Jones conjecture, the coarse Baum–Connes conjecture, and a description of higher order homological and homotopical Dehn functions. As a means of proving the Helly property we introduce and use the notion of a (generalized) cell Helly complex.

The braid group is a structure. It has the concepts and the implementations in graphics: points, lines, and some crosses. The crosses are also positive and negative. A braid not only forms graphs, but in different structures based on mathematics. So far, the braids only applies to discrete concepts, but it is possible to present a fuzzy concept. By applying computing to symbols representing the braids, the concept towards fuzzy as generating the form of another structure.

This paper studies quantum symmetric pairs ( U ~ , U ~ ı ) associated with quasi-split Satake diagrams of affine type A 2 r - 1 , D r , E 6 with a nontrivial diagram involution fixing the affine simple node. Various real and imaginary root vectors for the universal ı quantum groups U ~ ı are constructed with the help of the relative braid group action, and they are used to construct affine rank one subalgebras of U ~ ı . We then establish relations among real and imaginary root vectors in different affine rank one subalgebras and use them to give a Drinfeld type presentation of U ~ ı .

Let g be a symmetrisable Kac–Moody algebra and V an integrable g–module in category O. We show that the monodromy of the (normally ordered) rational Casimir connection on V can be made equivariant with respect to the Weyl group W of g, and therefore defines an action of the braid group BW on V. We then prove that this action is canonically equivalent to the quantum Weyl group action of BW on a quantum deformation of V, that is an integrable, category O module V over the quantum group Uħg such that V/ħV is isomorphic to V. This extends a result of the second author which is valid for g semisimple.

I report the recent advances in applying (graded) Hopf algebras with braided tensor product in two scenarios: i ) paraparticles beyond bosons and fermions living in any space dimensions and transforming under the permutation group; ii ) physical models of anyons living in two space-dimensions and transforming under the braid group. In the first scenario simple toy models based on the so-called 2-bit parastatistics show that, in the multiparticle sector, certain observables can discriminate paraparticles from ordinary bosons/fermions (thus, providing a counterexample to the widespread belief of the “conventionality of parastatistics” argument). In the second scenario the notion of (braided) Majorana qubit is introduced as the simplest building block to implement the Kitaev’s proposal of a topological quantum computer which protects from decoherence.

The challenge is to create an efficient quantum algorithm for the bosonic model capable of calculating the Jones polynomials for a knot resulting from interweaving or interlacing n -vertices. This weave is the construction of braid group representations from nineteen-vertex model. We present eigenbases and eigenvalues for lattice generators and their usefulness for the direct computation of Jones polynomials. The calculation shows that the Temperley-Lieb operators can be used for any braid word. Therefore, we propose a quantum sequence using these singular operators as quantum gates operating on the state of n qubits. We show that quantum calculations give the Jones polynomial for achiral knots and links.

Helical nanofibres play key roles in many biological processes. Entanglements between helices can aid gelation by producing thick, interconnected fibres, but the details of this process are poorly understood. Here, we describe the assembly of an achiral oligo(urea) peptidomimetic compound into supramolecular helices. Aggregation of adjacent helices leads to the formation of fibrils, which further intertwine to produce high-fidelity braids with periodic crossing patterns. A braid theory analysis suggests that braiding is governed by rigid topological constraints, and that branching occurs due to crossing defects in the developing braids. Mixed-chirality helices assemble into relatively complex, odd-stranded braids, but can also form helical bundles by undergoing inversions of chirality. The oligo(urea) assemblies are also highly sensitive to chiral amplification, proposed to occur through a majority-rules mechanism, whereby trace chiral materials can promote the formation of gels containing only homochiral helices. Helical structures play important roles in biological processes, yet their aggregation into fibres—which can in turn form gels—is poorly understood. Now, the self-assembly of a linear pentakis (urea) peptidomimetic compound into helices that further intertwine into well-defined braided structures has been described and analysed through braid theory. Homochiral gels may be formed by exposing the precursor sol to a chiral material.