Related papers: Matrix Characterization of Knots: A Simple Statist…
I briefly discuss a method of obtaining distinct classes of topologically equivalent knots by developing appropriate computer programs.
This is an introductory article on high dimensional knots for the beginners. High dimensional knot theory is an exciting field. It is a field of knot theory, which is one of topology and is connected with many ones. In this article we use…
We introduce a new way to tabulate knots by representing knot diagrams using a pair of planar trees. This pair of trees have their edges labeled by integers, they have no valence 2 vertices, and they have the same number of valence 1…
Knitting turns yarn, a 1D material, into a 2D fabric that is flexible, durable [1], and can be patterned to adopt a wide range of 3D geometries [2]. Like other mechanical metamaterials [3], the elasticity of knitted fabrics is an emergent…
A new computational technique based on the symbolic description utilizing kneading invariants is proposed and verified for explorations of dynamical and parametric chaos in a few exemplary systems with the Lorenz attractor. The technique…
We study the variation of the Tait number of a closed space curve according to its different projections. The results are used to compute the writhe of a knot, leading to a closed formula in case of polygonal curves.
In this paper, we give a precise and workable definition of a quantum knot system, the states of which are called quantum knots. This definition can be viewed as a blueprint for the construction of an actual physical quantum system.…
Quantum state space is endowed with a metric structure and Riemannian monotone metric is an important geometric entity defined on such a metric space. Riemannian monotone metrics are very useful for information-theoretic and statistical…
Knot theory is a study of the embedding of closed circles into three-dimensional Euclidean space, motivated the ubiquity of knots in daily life and human civilization. However, the current knot theory focuses on the topology rather than…
This paper presents a method for improved analysis of objects with an axial symmetry using X-ray Computed Tomography (CT). Cylindrical coordinates about an axis fixed to the object form the most natural base to check certain characteristics…
This paper is expository and is accessible to students. We define simple invariants of knots or links (linking number, Arf-Casson invariants and Alexander-Conway polynomials) motivated by interesting results whose statements are accessible…
Classical vector analysis is the predominant formalism used by engineers of computational electromagnetism, despite the fact that manifold as a theoretical concept has existed for a century. This paper discusses the benefits of manifolds…
A symmetric quandle is a quandle with a good involution. For a knot in \$R^3\$, a knotted surface in \$R^4\$ or an \$n\$-manifold knot in \$R^{n+2}\$, the knot symmetric quandle is defined. We introduce the notion of a symmetric quandle…
We study simple, knotted and linked torus windings that are made of tubes of finite thickness. Knots which have the shortest rope length are often denoted ideal structures. Conventionally, the ideal structure are found by rope shortening…
Symmetric extensions are essential in quantum mechanics, providing a lens to investigate the correlations of entangled quantum systems and to address challenges like the quantum marginal problem. Though semi-definite programming (SDP) is a…
The visibility graph of a simple polygon represents visibility relations between its vertices. Knowing the correct order of the vertices around the boundary of a polygon and its visibility graph, it is an open problem to locate the vertices…
We discuss a semi-implicit numerical scheme that allows for minimizing the bending energy of curves within certain isotopy classes. To this end we consider a weighted sum of the bending energy and the tangent-point functional. Based on…
We offer a pedestrian level review of the wall-crossing invariants. The story begins from the scattering theory in quantum mechanics where the spectrum reshuffling can be related to permutations of S-matrices. In non-trivial situations,…
While textiles have existed throughout much of human history as complex mechanical metamaterials, textile science has largely been overlooked by the physics community until recently. In this review, we consider the symmetry, topology, and…
In this review we establish various connections between complex networks and symmetry. While special types of symmetries (e.g., automorphisms) are studied in detail within discrete mathematics for particular classes of deterministic graphs,…