scholarly journals On torus knots and unknots

2016 ◽  
Vol 25 (06) ◽  
pp. 1650036 ◽  
Author(s):  
Chiara Oberti ◽  
Renzo L. Ricca
Keyword(s):  
Total Curvature ◽  
Topological Information ◽  
Torus Knots ◽  
Space Curves ◽  
Global Properties ◽  
Inflection Points ◽  
Total Squared Curvature ◽  
Curvature And Torsion ◽  
Total Torsion ◽  
Comprehensive Study

A comprehensive study of geometric and topological properties of torus knots and unknots is presented. Torus knots/unknots are particularly symmetric, closed, space curves, that wrap the surface of a mathematical torus a number of times in the longitudinal and meridian direction. By using a standard parametrization, new results on local and global properties are found. In particular, we demonstrate the existence of inflection points for a given critical aspect ratio, determine the location and prescribe the regularization condition to remove the local singularity associated with torsion. Since to first approximation total length grows linearly with the number of coils, its nondimensional counterpart is proportional to the topological crossing number of the knot type. We analyze several global geometric quantities, such as total curvature, writhing number, total torsion, and geometric ‘energies’ given by total squared curvature and torsion, in relation to knot complexity measured by the winding number. We conclude with a brief presentation of research topics, where geometric and topological information on torus knots/unknots finds useful application.

SPACE CURVES WITH NONZERO TORSION

1990 ◽  
Vol 01 (01) ◽  
pp. 109-117 ◽  
Author(s):  
BURT TOTARO
Keyword(s):  
Total Curvature ◽  
Closed Curve ◽  
Space Curves ◽  
Curvature And Torsion ◽  
Total Torsion ◽  
Nonzero Torsion

For a closed curve in R3 with curvature and torsion everywhere nonzero, the sum of the total curvature and the total torsion is greater than 4π.

TORUS KNOTS EMBODYING CURVATURE AND TORSION IN AN OTHERWISE FEATURELESS CONTINUUM

2011 ◽  
Vol 20 (12) ◽  
pp. 1723-1739 ◽  
Author(s):  
J. S. AVRIN
Keyword(s):  
Elementary Particle ◽  
General Relativity ◽  
Geometric Model ◽  
The Other ◽  
Particle Model ◽  
Torus Knots ◽  
Torus Knot ◽  
The Subject ◽  
Curvature And Torsion ◽  
The Relationship

The subject is a localized disturbance in the form of a torus knot of an otherwise featureless continuum. The knot's topologically quantized, self-sustaining nature emerges in an elementary, straightforward way on the basis of a simple geometric model, one that constrains the differential geometric basis it otherwise shares with General Relativity (GR). Two approaches are employed to generate the knot's solitonic nature, one emphasizing basic differential geometry and the other based on a Lagrangian. The relationship to GR is also examined, especially in terms of the formulation of an energy density for the Lagrangian. The emergent knot formalism is used to derive estimates of some measurable quantities for a certain elementary particle model documented in previous publications. Also emerging is the compatibility of the torus knot formalism and, by extension, that of the cited particle model, with general relativity as well as with the Dirac theoretic notion of antiparticles.

scholarly journals Total Curvature and Total Torsion of Knotted Polymers

2007 ◽  
Vol 40 (10) ◽  
pp. 3860-3867 ◽  
Author(s):  
Patrick Plunkett ◽  
Michael Piatek ◽  
Akos Dobay ◽  
John C. Kern ◽  
Kenneth C. Millett ◽  
...  
Keyword(s):  
Total Curvature ◽  
Total Torsion

The Image of the Coupler Motion of a Special Spherical Four Bar Linkage

1987 ◽  
Author(s):  
J. M. McCarthy
Keyword(s):  
Rational Functions ◽  
Global Properties ◽  
Kinematic Mapping ◽  
Curvature And Torsion ◽  
Four Bar Linkage ◽  
Spherical Motion

Abstract This paper uses a kinematic mapping of spherical motion to derive an image curve which represents the coupler motion of a doubly folding spherical four bar linkage. The image curve of this linkage, the so called “kite” linkage, can be parameterized by rational functions. This parameterization is presented as well as formulas which allow the computation of its curvature and torsion at any point. These formulas provide a link between the global properties of the coupler motion as represented by the image curve itself and its instantaneous properties given by the curvature and torsion functions.

TWIST SEQUENCES AND VASSILIEV INVARIANTS

1994 ◽  
Vol 03 (03) ◽  
pp. 391-405 ◽  
Author(s):  
ROLLAND TRAPP
Keyword(s):  
Finite Type ◽  
Polynomial Growth ◽  
Difference Sequence ◽  
Uniform Limit ◽  
Topological Information ◽  
Torus Knots ◽  
Knot Invariants ◽  
Vassiliev Invariants ◽  
Finite Type Invariants ◽  
Vassiliev Invariant

In this paper we describe a difference sequence technique, hereafter referred to as the twist sequence technique, for studying Vassiliev invariants. This technique is used to show that Vassiliev invariants have polynomial growth on certain sequences of knots. Restrictions of Vassiliev invariants to the sequence of (2, 2i + 1) torus knots are characterized. As a corollary it is shown that genus, crossing number, signature, and unknotting number are not Vassiliev invariants. This characterization also determines the topological information about (2, 2i + 1) torus knots encoded in finite-type invariants. The main result obtained is that the complement of the space of Vassiliev invariants is dense in the space of all numeric knot invariants. Finally, we show that the uniform limit of a sequence of Vassiliev invariants must be a Vassiliev invariant.

scholarly journals ESTIMATING TORSION OF DIGITAL CURVES USING 3D IMAGE ANALYSIS

2016 ◽  
Vol 35 (2) ◽  
pp. 81 ◽  
Author(s):  
Christoph Blankenburg ◽  
Christian Daul ◽  
Joachim Ohser
Keyword(s):  
A Priori ◽  
Three Dimensional ◽  
Synthetic Data ◽  
3D Image ◽  
Continuous Curve ◽  
Space Curves ◽  
Discrete Curves ◽  
Curvature Estimation ◽  
Tomographic Data ◽  
Curvature And Torsion

Curvature and torsion of three-dimensional curves are important quantities in fields like material science or biomedical engineering. Torsion has an exact definition in the continuous domain. However, in the discrete case most of the existing torsion evaluation methods lead to inaccurate values, especially for low resolution data. In this contribution we use the discrete points of space curves to determine the Fourier series coefficients which allow for representing the underlying continuous curve with Cesàro’s mean. This representation of the curve suits for the estimation of curvature and torsion values with their classical continuous definition. In comparison with the literature, one major advantage of this approach is that no a priori knowledge about the shape of the cyclic curve parts approximating the discrete curves is required. Synthetic data, i.e. curves with known curvature and torsion, are used to quantify the inherent algorithm accuracy for torsion and curvature estimation. The algorithm is also tested on tomographic data of fiber structures and open foams, where discrete curves are extracted from the pore spaces.

The Image Curve of the Coupler of a Special Spherical Four Bar Linkage

1988 ◽  
Vol 110 (3) ◽  
pp. 276-280
Author(s):  
J. M. McCarthy
Keyword(s):  
Rational Functions ◽  
Global Properties ◽  
Kinematic Mapping ◽  
Curvature And Torsion ◽  
Four Bar Linkage ◽  
Spherical Motion

This paper uses a kinematic mapping of spherical motion to derive an image curve which represents the coupler motion of a doubly folding spherical four bar linkage. The image curve of this linkage, the so called “kite” linkage, can be parameterized by rational functions. This parameterization is presented as well as formulas which allow the computation of its curvature and torsion at any point. These formulas provide a link between the global properties of the coupler motion as represented by the image curve itself and its instantaneous properties given by the curvature and torsion functions.

Gridding with continuous curvature splines in tension

Geophysics ◽  
1990 ◽  
Vol 55 (3) ◽  
pp. 293-305 ◽  
Author(s):  
W. H. F. Smith ◽  
P. Wessel
Keyword(s):  
Elastic Plate ◽  
General System ◽  
Computational Effort ◽  
Tension Parameter ◽  
Plate Flexure ◽  
Inflection Points ◽  
Method Of Solution ◽  
Total Squared Curvature ◽  
Pass Through ◽  
Minimum Curvature

A gridding method commonly called minimum curvature is widely used in the earth sciences. The method interpolates the data to be gridded with a surface having continuous second derivatives and minimal total squared curvature. The minimum‐curvature surface has an analogy in elastic plate flexure and approximates the shape adopted by a thin plate flexed to pass through the data points. Minimum‐curvature surfaces may have large oscillations and extraneous inflection points which make them unsuitable for gridding in many of the applications where they are commonly used. These extraneous inflection points can be eliminated by adding tension to the elastic‐plate flexure equation. It is straightforward to generalize minimum‐curvature gridding algorithms to include a tension parameter; the same system of equations must be solved in either case and only the relative weights of the coefficients change. Therefore, solutions under tension require no more computational effort than minimum‐curvature solutions, and any algorithm which can solve the minimum‐curvature equations can solve the more general system. We give common geologic examples where minimum‐curvature gridding produces erroneous results but gridding with tension yields a good solution. We also outline how to improve the convergence of an iterative method of solution for the gridding equations.

scholarly journals Symmetry detection of rational space curves from their curvature and torsion

2015 ◽  
Vol 33 ◽  
pp. 51-65 ◽  
Author(s):  
Juan Gerardo Alcázar ◽  
Carlos Hermoso ◽  
Georg Muntingh
Keyword(s):  
Symmetry Detection ◽  
Rational Space ◽  
Space Curves ◽  
Curvature And Torsion

Rail Geometry and Euler Angles

2006 ◽  
Vol 1 (3) ◽  
pp. 264-268 ◽  
Author(s):  
Cheta Rathod ◽  
Ahmed A. Shabana
Keyword(s):  
Vehicle Dynamics ◽  
Euler Angles ◽  
Frenet Frame ◽  
Geometric Properties ◽  
Track Geometry ◽  
Railroad Vehicle Dynamics ◽  
Space Curves ◽  
Curvature And Torsion ◽  
The Relationship ◽  
Railroad Vehicle

In railroad vehicle dynamics, Euler angles are often used to describe the track geometry (track centerline and rail space curves). The tangent and curvature vectors as well as local geometric properties such as the curvature and torsion can be expressed in terms of Euler angles. Some of the local geometric properties and Euler angles can be related to measured parameters that are often used to define the track geometry. The Euler angles employed, however, define a coordinate system that may differ from the Frenet frame used in the classical differential geometry. The relationship between the track frame used in railroad vehicle dynamics and the Frenet frame used in the theory of curves is developed in this paper and is used to shed light on some of the formulas and identities used in the geometric description in railroad vehicle dynamics. The conditions under which the two frames (track and Frenet) become equivalent are presented and used to obtain expressions for the curvature and torsion in terms of Euler angles and their derivatives with respect to the arc length.

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