This research paper aims to study a type of geometric decorations that prevailed in all Islamic arts in Iran and some regions of Central Asia, especially Khorasan Province, from the 10th to the 13th ...G centuries. Then, it spread through the Islamic world and was found in the Ayyubid and Mamluk arts in Egypt and the arts of Morocco and Andalusia. The paper also explores the reasons for the emergence, relation to place, and symbolism of this decoration at the time. It highlights the origin of this decoration. The arts of the eastern area of the Islamic World experienced the prevalence of geometric decorations, including all types of knots on their products. The present paper explores the "knot of good luck" with its various forms known as decorative items in the ancient era and many cultures. It discusses this decoration's several names and symbolism. It highlights the causes of using these knots in the Islamic arts of that period and corrects some misconceptions about the emergence of this knot.
The determination of the dynamic contact angle is of significant interest for the characterization of the wettability of technical fibers and textiles in diverse fields of science and technology. ...There exist traditional methods for dynamic contact angle measurements of flat surfaces and of fibers with a uniform cross‐sectional shape along the fiber. So far, however, no method has been reported which is suitable for structured fibers, particularly for spindle‐knotted structured fibers of varying cross‐sections. This article describes a new method for measuring the dynamic contact angle for polydimethylsiloxane (PDMS) spindle‐knotted structured fibers. The method is an outcome of integrating the results obtained from experiments (applying force tensiometry) and a proposed theoretical model describing such fibers. The reliability and conformity of the results are shown by comparing the measured dynamic contacts angle of PDMS as spindle‐knot and as a flat surface. This method may pave the road for better wettability analysis of various structured fibers. It also allows to measure the local receding and advancing contact angles for macroscopic/microscopic structured fibers (especially when they are not accessible as flat surfaces) against the various test liquids.
A method is developed to measure the dynamic contact angle of PDMS spindle‐knotted structured fiber. The method is an outcome of integrating results from the experiment (force tensiometry) and the theoretical model based on the position‐dependent forces acting on the fiber, taking into account its non‐uniform structure. Generalizing the method is straightforward for other structured fibers to analyze their wettability.
Plasma knots Gross, Oliver; Pinkall, Ulrich; Schröder, Peter
Physics letters. A,
08/2023, Letnik:
480
Journal Article
Recenzirano
We present a Lagrangian method for the computation of ideal plasma knots and links. It is based on a variational principle for stable equilibria of an ideal plasma in the case of a free boundary ...subjected to external magnetic or plasma pressure forces. For this purpose, we introduce a structure preserving discretization of plasma based on decompositions of Riemannian manifolds representing pressure confined plasma regions in magnetohydrostatic equilibrium. Moreover, we show that, by the virtue of an analogy, the method can be used for the approximation of steady Euler-flows of arbitrarily complex topology.
•A Lagrangian method for computing ideal plasma knots and links.•A variational principle for ideal plasma with a free boundary subject to external magnetic or plasma pressure forces.•A structure preserving discretization of ideal plasma.•The method can be used for approximating steady Euler-flows of arbitrarily complex topology.
This article reports the results of an investigation into the average behavior of the knot spectrum (of knots up to 16 crossings) of a family of random knot spaces. A knot space in this family ...consists of random polygons of a given length in a spherical confinement of a given radius. The knot spectrum is the distribution of all knot types within a random knot space and is based on the probabilities that a randomly (and uniformly) chosen polygon from this knot space forms different knot types. We show that the relative spectrum of knots, when divided into groups by their crossing number, remains unexpectedly robust as these knot spaces vary. The relative spectrum for a given crossing number c is P
c
(u)/P
c
, where P
c
is the probability that a uniformly chosen random polygon has crossing number c, and P
c
(u) is the probability that the chosen polygon has crossing number c and is from the group of knots defined by the characteristic u (such as "alternating prime," "nonalternating prime," or "composite"). Specifically, for a fixed crossing number c, the results show that tighter confinement conditions favor alternating prime knots, that is P
c
(A)/P
c
(where A stands for "alternating prime") increases as the confinement radius decreases, and that the average P
c
(N) (where N stands for "nonalternating prime") behaves similar to the average P
c − 1
(A). We then use our simulations to speculate on limiting behavior.
We consider free symmetries on cobordisms between knots, which is equivalent to cobordisms between knots in lens spaces. We classify which freely periodic knots bound equivariant surfaces in the ...4-ball in terms of corresponding homology classes in lens spaces. We give a numerical condition determining the free periods for which torus knots bound equivariant surfaces in the 4-ball.
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.
In this paper, we discuss a nonuniform extension of the Möbius energy for curves of nonuniform thickness. After defining the nonuniform energy and stating its basic properties, we show that its C1 ...minimizers in R3 must be convex planar curves. As this energy is not Möbius invariant, this requires a novel approach. We then describe two variations of the nonuniform energy: the equitotal variation, which allows the free reallocation of weight along a curve, and the translatory variation, which only allows the translation of a weight profile along a curve. We prove that equitotal variation has no absolute minimizer when the curve has a point of zero curvature. We also provide conditions that ensure invariance of the energy of a weighted curve under translatory variation—a standard parametrization of torus knots with sinusoidal weight distributions are such examples.
An invariant for colored bonded knots Gabrovšek, Boštjan
Studies in applied mathematics (Cambridge),
April 2021, 2021-04-00, 20210401, Letnik:
146, Številka:
3
Journal Article
Recenzirano
We equip a knot K with a set of colored bonds, that is, colored intervals properly embedded into R3∖K. Such a construction can be viewed as a structure that topologically models a closed protein ...chain including any type of bridges connecting the backbone residues. We introduce an invariant of such colored bonded knots that respects the HOMFLYPT relation, namely, the HOMFLYPT skein module of colored bonded knots. We show that the rigid version of the module is freely generated by colored Θ‐curves and handcuff links, while the nonrigid version is freely generated by the trivially embedded Θ‐curve. The latter module, however, does not provide information about the knottedness of the bonds.
Entangling strands in a well-ordered manner can produce useful effects, from shoelaces and fishing nets to brown paper packages tied up with strings. At the nanoscale, non-crystalline polymer chains ...of sufficient length and flexibility randomly form tangled mixtures containing open knots of different sizes, shapes and complexity. However, discrete molecular knots of precise topology can also be obtained by controlling the number, sequence and stereochemistry of strand crossings: orderly molecular entanglements. During the last decade, substantial progress in the nascent field of molecular nanotopology has been made, with general synthetic strategies and new knotting motifs introduced, along with insights into the properties and functions of ordered tangle sequences. Conformational restrictions imparted by knotting can induce allostery, strong and selective anion binding, catalytic activity, lead to effective chiral expression across length scales, binding modes in conformations efficacious for drug delivery, and facilitate mechanical function at the molecular level. As complex molecular topologies become increasingly synthetically accessible they have the potential to play a significant role in molecular and materials design strategies. We highlight particular examples of molecular knots to illustrate why these are a few of our favourite things.
We review recent progress in molecular knotting, the chemistry of orderly molecular entanglements. As complex nanotopologies become increasingly accessible they may play significant roles in molecular design.
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