THE SELF-LINKING NUMBER OF A CLOSED CURVE IN ℝn

2000 ◽  
Vol 09 (04) ◽  
pp. 491-503
Author(s):  
A. MONTESINOS AMILIBIA ◽  
J. J. NUÑO BALLESTEROS
Keyword(s):  
Vector Bundle ◽  
Closed Curve ◽  
The Self ◽  
Regularity Conditions ◽  
Dimensional Vector ◽  
3 Dimensional ◽  
Orthogonal Vector ◽  
Linking Number

We introduce the self-linking number of a smooth closed curve α:S1→ℝn with respect to a 3-dimensional vector bundle over the curve, provided that some regularity conditions are satisfied. When n=3, this construction gives the classical self-linking number of a closed embedded curve with non-vanishing curvature [5]. We also look at some interesting particular cases, which correspond to the osculating or the orthogonal vector bundle of the curve.

scholarly journals LINKING NUMBER IN A PROJECTIVE SPACE AS THE DEGREE OF A MAP

2007 ◽  
Vol 16 (04) ◽  
pp. 489-497 ◽  
Author(s):  
JULIA VIRO DROBOTUKHINA
Keyword(s):  
Projective Space ◽  
Configuration Space ◽  
The Self ◽  
Real Projective Space ◽  
3 Dimensional ◽  
Linking Number ◽  
Degree Of A Map ◽  
Number Of Cycles ◽  
Dimensional Projective Space

For any two disjoint oriented circles embedded into the 3-dimensional real projective space, we construct a 3-dimensional configuration space and its map to the projective space such that the linking number of the circles is the half of the degree of the map. Similar interpretations are given for the linking number of cycles in a projective space of arbitrary odd dimension and the self-linking number of a zero homologous knot in the 3-dimensional projective space.

scholarly journals New Formulas and Results for 3-Dimensional Vector Fields

2021 ◽  
Vol 12 (11) ◽  
pp. 1058-1096
Author(s):  
Sadanand D. Agashe
Keyword(s):  
Vector Fields ◽  
Dimensional Vector ◽  
3 Dimensional

Bed forms in turbulent channel flow

2004 ◽  
Vol 57 (1) ◽  
pp. 77-93 ◽  
Author(s):  
Albert Gyr and ◽  
Wolfgang Kinzelbach
Keyword(s):  
Orange Peel ◽  
Turbulent Channel Flow ◽  
Secondary Flows ◽  
The Self ◽  
Self Organization ◽  
3 Dimensional ◽  
Bed Forms ◽  
Bed Form ◽  
Separation Zones ◽  
Grain Diameter

Bed forms in channels result from the interaction between sediment transport, turbulence and gravitational settling. They document mechanisms of self-organization between flow structures and the developing structure of the bed. It is shown that these mechanisms can be characterized by length scales of the sediment, the bed form and the flow structure. Three types of interaction can be distinguished: 1) The first type of mechanisms can be observed at beds of sediment with grain diameter smaller than the typical structural dimension of turbulence. It is shown how with increasing hydraulic loading of the bed a hydraulically smooth surface develops structures, which turn from “orange peel” to stripe and arrowhead patterns and finally into ripples. This group of bed forms is limited to a grain diameter of d+=12.5 in viscous units. In the regime of the stripe structures drag reduction occurs. 2) If grains or bed forms reach a height, which leads to separation, a completely different regime prevails, which is determined by the self-organization of separation zones. A prominent example for these bed forms are dunes. 3) Demixing processes, secondary flows and roughness contrasts finally lead to the development of longitudinal and transverse banks. All three mechanisms are explained on the basis of kinematic models and documented by experimental data. Emphasis is put on the two-dimensionalization of bed forms in a highly 3-dimensional (3D) turbulent flow, which is traced back to the self organization of vortex systems. This review article contains 55 references.

scholarly journals The number of framings of a knot in a 3-manifold

2014 ◽  
Vol 23 (13) ◽  
pp. 1450072 ◽  
Author(s):  
Patricia Cahn ◽  
Vladimir Chernov ◽  
Rustam Sadykov
Keyword(s):  
Normal Bundle ◽  
Closed Curve ◽  
The Self ◽  
Connected Sum ◽  
Orientable Manifold ◽  
Intersection Points

In view of the self-linking invariant, the number |K| of framed knots in S3 with given underlying knot K is infinite. In fact, the second author previously defined affine self-linking invariants and used them to show that |K| is infinite for every knot in an orientable manifold unless the manifold contains a connected sum factor of S1 × S2; the knot K need not be zero-homologous and the manifold is not required to be compact. We show that when M is orientable, the number |K| is infinite unless K intersects a nonseparating sphere at exactly one point, in which case |K| = 2; the existence of a nonseparating sphere implies that M contains a connected sum factor of S1 × S2. For knots in nonorientable manifolds we show that if |K| is finite, then K is disorienting, or there is an orientation-preserving isotopy of the knot to itself which changes the orientation of its normal bundle, or it intersects some embedded S2 or ℝP2 at exactly one point, or it intersects some embedded S2 at exactly two points in such a way that a closed curve consisting of an arc in K between the intersection points and an arc in S2 is disorienting.

scholarly journals Quaternions and hybrid nematic disclinations

2013 ◽  
Vol 469 (2156) ◽  
pp. 20130204 ◽  
Author(s):  
Simon Čopar ◽  
Slobodan Žumer
Keyword(s):  
Liquid Crystals ◽  
Homotopy Theory ◽  
Classification Scheme ◽  
Colloidal Suspensions ◽  
The Self ◽  
Topological Classification ◽  
Winding Number ◽  
Linking Number ◽  
Disclination Line ◽  
Geometric Decomposition

Disclination lines in nematic liquid crystals can exist in different geometric conformations, characterized by their director profile. In certain confined colloidal suspensions and even more prominently in chiral nematics, the director profile may vary along the disclination line. We construct a robust geometric decomposition of director profile in closed disclination loops and use it to apply topological classification to linked loops with arbitrary variation of the profile, generalizing the self-linking number description of disclination loops with the winding number . The description bridges the gap between the known abstract classification scheme derived from homotopy theory and the observable local features of disclinations, allowing application of said theory to structures that occur in practice.

Fiducial Registration With Spoiled Gradient-Echo Magnetic Resonance Imaging Enhances the Accuracy of Subthalamic Nucleus Targeting

Neurosurgery ◽  
2011 ◽  
Vol 69 (4) ◽  
pp. 870-875 ◽  
Author(s):  
Sharona Ben-Haim ◽  
Yakov Gologorsky ◽  
Ann Monahan ◽  
Donald Weisz ◽  
Ron L Alterman
Keyword(s):  
Dimensional Vector ◽  
Microelectrode Recording ◽  
Resonance Imaging ◽  
Gradient Echo ◽  
3 Dimensional ◽  
Vector Difference ◽  
Group 2 ◽  
Spoiled Gradient Echo ◽  
Deep Brain ◽  
Group 1

Abstract BACKGROUND: A variety of imaging strategies may be used to derive reliable stereotactic coordinates when performing deep brain stimulation lead implants. No single technique has yet proved optimal. OBJECTIVE: To compare the relative accuracy of stereotactic coordinates for the subthalamic nucleus (STN) derived either from fast spin echo/inversion recovery (FSE/IR) magnetic resonance imaging MRI alone (group 1) or FSE/IR in conjunction with T1-weighted spoiled gradient-echo MRI (group 2). METHODS: A retrospective analysis of 145 consecutive STN deep brain stimulation lead placements (group 1, n = 72; group 2, n = 73) was performed in 81 Parkinson disease patients by 1 surgical team. From the operative reports, we recorded the number of microelectrode recording trajectories required to localize the desired STN target and the span of STN traversed along the implantation trajectory. In addition, we calculated the 3-dimensional vector difference between the initial MRI-derived coordinates and the final physiologically refined coordinates. RESULTS: The proportion of implants completed with just 1 microelectrode recording trajectory was greater (81% vs 58%; P < .001) and the 3-dimensional vector difference between the anatomically selected target and the microelectrode recording–refined target was smaller (0.6 ± 1.2 vs 0.9 ± 1.3; P = .04) in group 2 than in group 1. At the same time, the mean expanse of STN recorded along the implantation trajectory was 8% greater in group 2 (4.8 ± 0.6 vs 5.2 ± 0.6 mm; P < .001). CONCLUSION: A combination of stereotactic FSE/IR and spoiled gradient-echo MRI yields more accurate coordinates for the STN than FSE/IR MRI alone.

Sign in / Sign up

Export Citation Format

Share Document