Related papers: Weaving knotted vector fields with tunable helicit…
It is shown that every knot or link is the set of complex tangents of a 3-sphere smoothly embedded in the three-dimensional complex space. We show in fact that a one-dimensional submanifold of a closed orientable 3-manifold can be realised…
We establish two spectral sequences in knot Floer homology associated to a directed strongly invertible knot K: one from the knot Floer homology of K to a two dimensional vector space, and one from the singular knot Floer homology of a…
Let $\phi : S^1\times D^2\to S^1$ be the natural projection. An oriented knot $K\hookrightarrow V = S^1\times D^2$ is called an almost closed braid if the restriction of $\phi$ to K has exactly two (non-degenerate) critical points (and K is…
In this paper we outline a topological framework for constructing 2-periodic knitted stitches and an algebra for joining stitches together to form more complicated textiles. Our topological framework can be constructed from certain…
The Skyrme-Faddeev model is a three-dimensional non-linear field theory that has topological soliton solutions, called hopfions, which are novel string-like solutions taking the form of knots and links. Solutions found thus far take the…
We define an elementary relatively $\mathbb Z/4$ graded Lagrangian-Floer chain complex for restricted immersions of compact 1-manifolds into the pillowcase, and apply it to the intersection diagram obtained by taking traceless $SU(2)$…
In last 50 years, a significant progress was noticed in medicine, communications and entertainment. Such advanced development of these fields was directly related to ability of controlling light. Photonics is exactly about this ability. At…
Representing vector fields by potentials can be a challenging task in domains with cavities or tunnels, due to the presence of harmonic fields which are both irrotational and solenoidal but may have no scalar or vector potentials. For…
We show that one can express the knot equation of Skyrme theory completely in terms of the vacuum potential of SU(2) QCD, in such a way that the equation is viewed as a generalized Lorentz gauge condition which selects one vacuum for each…
Torse-forming vector fields are generalizations of some important vector fields. In this paper, we present some techniques to transform a proper torse-forming vector field into its special cases. Concrete examples are given.
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…
We construct meta-stable knotted domain strings on the surface of a soliton of the shape of a torus in 3+1 dimensions. We consider the simplest case of Z2 Wess-Zumino-type domain walls for which we can cover the torus with a domain string…
New results on the kinetic energy of ideal vortex filaments in the shape of torus knots and unknots are presented. These knots are given by small-amplitude torus knot solutions (Ricca, 1993) to the Localized Induction Approximation (LIA)…
We report on the geometry and mechanics of knotted stiff strings. We discuss both closed and open knots. Our two main results are: (i) Their equilibrium energy as well as the equilibrium tension for open knots depend on the type of knot as…
Mechanical metamaterials have continued to offer unprecedented tunability in mechanical properties, but most designs to date have prioritized attaining high stiffness and strength while sacrificing deformability. The emergence of woven…
Vector beams are often regarded as non-separable superpositions of spatial and polarization degrees of freedom that satisfy the wave equation. This interpretation ties their polarization structure to their spatial shape. Here, we introduce…
Weaving knots are alternating knots with the same projection as torus knots, and were conjectured by X.-S. Lin to be among the maximum volume knots for fixed crossing number. We provide the first asymptotically correct volume bounds for…
We numerically study the evolution of a small turbulent region of quantised vorticity in superfluid helium, a regime which can be realised in the laboratory. We show that the turbulence achieves a fluctuating steady-state in terms of…
We describe a general procedure for mapping arbitrary $n$-qubit states to two-dimensional (2D) vector fields. The mappings use complex rational function representations of individual qubits, producing classical vector field configurations…
We present a universal method to create a tunable, artificial vector gauge potential for neutral particles trapped in an optical lattice. The necessary Peierls phase of the hopping parameters between neighboring lattice sites is generated…