An index formula for the self-linking number of a space curve

2008 ◽  
Vol 134 (1) ◽  
pp. 197-202 ◽  
Author(s):  
Peter Røgen
Keyword(s):  
Space Curve ◽  
The Self ◽  
Index Formula ◽  
Linking Number

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.

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.

FRAMED KNOTS IN 3-MANIFOLDS AND AFFINE SELF-LINKING NUMBERS

2005 ◽  
Vol 14 (06) ◽  
pp. 791-818 ◽  
Author(s):  
VLADIMIR CHERNOV TCHERNOV
Keyword(s):  
The Self ◽  
Compact Manifolds ◽  
Not Given ◽  
Connected Sum ◽  
Linking Number ◽  
Fundamental Fact ◽  
Linking Numbers

The number |K| of non-isotopic framed knots that correspond to a given unframed knot K ⊂ S3 is infinite. This follows from the existence of the self-linking number slk of a zero homologous framed knot. We use the approach of Vassiliev–Goussarov invariants to construct "affine self-linking numbers" that are extensions of slk to the case of nonzero homologous framed knots in 3-manifolds. As a corollary we get that |K| = ∞ for all knots in an oriented (not necessarily compact) 3-manifold M that is not realizable as a connected sum (S1 × S2)# M′. This result for compact manifolds was first stated by Hoste and Przytycki. They referred to the works of McCullough for the idea of the proof, however to the best of our knowledge prior to this work the proof of this fundamental fact was not given in literature or in a preprint form. Our proof is based on different ideas. For M = (S1 × S2)# M′ we construct K in M such that |K| = 2 ≠ ∞.

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 Topological Indices of Proteins

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
Dmitry Melnikov ◽  
Antti J. Niemi ◽  
Ara Sedrakyan
Keyword(s):  
Topological Indices ◽  
The Self ◽  
Topological Classification ◽  
Discrete Version ◽  
Linking Number ◽  
Inflection Points ◽  
Folding Dynamics ◽  
Polygonal Chains ◽  
Special Points

Abstract Protein molecules can be approximated by discrete polygonal chains of amino acids. Standard topological tools can be applied to the smoothening of the polygons to introduce a topological classification of folded states of proteins, for example, using the self-linking number of the corresponding framed curves. In this paper we extend this classification to the discrete version, taking advantage of the “randomness” of such curves. Known definitions of the self-linking number apply to non-singular framings: for example, the Frenet framing cannot be used if the curve has inflection points. However, in the discrete proteins the special points are naturally resolved. Consequently, a separate integer topological characteristics can be introduced, which takes into account the intrinsic features of the special points. This works well for the proteins in our analysis, for which we compute integer topological indices associated with the singularities of the Frenet framing. We show how a version of the Calugareanu’s theorem is satisfied for the associated self-linking number of a discrete curve. Since the singularities of the Frenet framing correspond to the structural motifs of proteins, we propose topological indices as a technical tool for the description of the folding dynamics of proteins.

A CONSTRUCTION OF TRANSVERSELY NON-SIMPLE KNOT TYPES

2012 ◽  
Vol 21 (11) ◽  
pp. 1250108
Author(s):  
HIROSHI MATSUDA
Keyword(s):  
The Self ◽  
Linking Number ◽  
Transverse Knots ◽  
Standard Contact

For n = 1, 2 and 3, we construct a pair of transverse knots T1 and [Formula: see text], in the standard contact 3-sphere, satisfying the following properties: (1) the topological knot type of T1 is the same as that of [Formula: see text], (2) the self-linking number of T1 is equal to that of [Formula: see text], (3) [Formula: see text] is obtained from a transverse knot T2 by n stabilizations, and (4) T1 is not transversely isotopic to [Formula: see text].

scholarly journals Geometry of Călugăreanu's theorem

2005 ◽  
Vol 461 (2062) ◽  
pp. 3245-3254 ◽  
Author(s):  
M.R Dennis ◽  
J.H Hannay
Keyword(s):  
Crossing Number ◽  
Space Curve ◽  
Closed Space ◽  
Linking Number ◽  
Space Geometry ◽  
The Common ◽  
Central Result

A central result in the space geometry of closed twisted ribbons is Călugăreanu's theorem (also known as White's formula, or the Călugăreanu–White–Fuller theorem). This enables the integer linking number of the two edges of the ribbon to be written as the sum of the ribbon twist (the rate of rotation of the ribbon about its axis) and its writhe. We show that twice the twist is the average, over all projection directions, of the number of places where the ribbon appears edge-on (signed appropriately)—the ‘local’ crossing number of the ribbon edges. This complements the common interpretation of writhe as the average number of signed self-crossings of the ribbon axis curve. Using the formalism we develop, we also construct a geometrically natural ribbon on any closed space curve—the ‘writhe framing’ ribbon. By definition, the twist of this ribbon compensates its writhe, so its linking number is always zero.

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