GEOMETRIC KNOT SPACES AND POLYGONAL ISOTOPY

The space of n-sided polygons embedded in three-space consists of a smooth manifold in which points correspond to piecewise linear or "geometric" knots, while paths correspond to isotopies which preserve the geometric structure of these knots. The topology of these spaces for the case n=6 and n=7 is described. In both of these cases, each knot space consists of five components,but contains only three (when n=6) or four (when n=7) topological knot types. Therefore "geometric knot equivalence" is strictly stronger than topological equivalence. This point is demonstrated by the hexagonal trefoils and heptagonal figure-eight knots, which, unlike their topological counterparts, are not reversible. Extending these results to the cases n≥8 will also be discussed.

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On geometric structure of phase portraits of some piecewise linear dynamical systems

TBILISI - MATHEMATICS ◽

10.2478/9788366675476-004 ◽

2021 ◽

pp. 49-56

Author(s):

V. P. Golubyatnikov ◽

L. S. Minushkina

Keyword(s):

Dynamical Systems ◽

Geometric Structure ◽

Piecewise Linear ◽

Phase Portraits ◽

Linear Dynamical Systems

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CLIFFORD AND EXTENSOR CALCULUS AND THE RIEMANN AND RICCI EXTENSOR FIELDS OF DEFORMED STRUCTURES (M, ∇′, η) AND (M, ∇, g)

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s021988780700248x ◽

2007 ◽

Vol 04 (07) ◽

pp. 1159-1172 ◽

Cited By ~ 4

Author(s):

V. V. FERNÁNDEZ ◽

W. A. RODRIGUES ◽

A. M. MOYA ◽

R. DA ROCHA

Keyword(s):

Differential Geometry ◽

Gravitational Field ◽

Geometric Structure ◽

Smooth Manifold ◽

Important Case ◽

Geometrical Structures ◽

Open Set ◽

Geometric Algebras ◽

General Connection

Here (the last paper in a series of four) we end our presentation of the basics of a systematical approach to the differential geometry of a smooth manifold M (supporting a metric field g and a general connection ∇) which uses the geometric algebras of multivector and extensors (fields) developed in previous papers. The theory of the Riemann and Ricci fields of a triple (M, ∇, g) is investigated for each particular open set U ⊂ M through the introduction of a geometric structure on U, i.e. a triple (U, γ, g), where γ is a general connection field on U and g is a metric extensor field associated to g. The relation between geometrical structures related to gauge extensor fields is clarified. These geometries may be said to be deformations one of each other. Moreover, we study the important case of a class of deformed Levi–Civita geometrical structures and prove key theorems about them that are important in the formulation of geometric theories of the gravitational field.

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ON SMOOTHING PROPER KNOTS IN 3-MANIFOLDS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216503002895 ◽

2003 ◽

Vol 12 (07) ◽

pp. 971-985 ◽

Cited By ~ 1

Author(s):

OLLIE NANYES

Keyword(s):

Real Line ◽

Piecewise Linear ◽

Sufficient Conditions ◽

Topological Equivalence ◽

The Real

A topological proper knot is a proper embedding f:ℝ1→M3 of the real line into an open 3-manifold. Two proper knots are equivalent if they can be connected by a topological proper isotopy. In this paper, we answer a question posed by the author in [6] and show that, up to topological equivalence and orientation, all proper knots running between the opposite ends of D2×ℝ1 are equivalent. Then sufficient conditions for a topological proper knot to be equivalent to a piecewise linear proper knot are given.

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CHUA'S OSCILLATOR: A ZOO OF ATTRACTORS

Journal of Circuits System and Computers ◽

10.1142/s0218126693000228 ◽

1993 ◽

Vol 03 (02) ◽

pp. 309-359 ◽

Cited By ~ 14

Author(s):

PHILIPPE DEREGEL

Keyword(s):

Vector Fields ◽

Piecewise Linear ◽

Topological Equivalence ◽

Chua’S Circuit ◽

Explicit Formulas ◽

Chua's Circuit ◽

Chua’S Oscillator ◽

In Series ◽

User Friendly ◽

Circuit Family

Chaos has been widely reported and studied in Chua's circuit family, which is characterized by a 21 parameter family of odd-symmetric piecewise-linear vector fields in R3. In this tutorial paper, we shall prove that, up to a topological equivalence, all the dynamics of this family are subsumed within that of a single circuit: Chua's oscillator; directly derived from Chua's circuit by adding a resistor in series with the inductor. We provide explicit formulas of the parameters of Chua's oscillator leading to a behavior qualitatively identical to that of any system belonging to Chua's circuit family. These formulas are then used to construct, in an almost trivial way, a gallery of (quasiperiodic and strange) attractors belonging to Chua's circuit family. A user-friendly program is available to allow a better understanding of the evolution of the dynamics as a function of the parameters of Chua's oscillator, and to follow the trajectory in the eigenspaces.

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Parabolic conformally symplectic structures I; definition and distinguished connections

Forum Mathematicum ◽

10.1515/forum-2017-0018 ◽

2018 ◽

Vol 30 (3) ◽

pp. 733-751 ◽

Cited By ~ 3

Author(s):

Andreas Čap ◽

Tomáš Salač

Keyword(s):

Lie Algebra ◽

Geometric Structure ◽

Tangent Bundle ◽

Smooth Manifold ◽

Symplectic Structure ◽

Linear Connection ◽

Underlying Structure ◽

Normalization Condition ◽

Lie Algebra Cohomology ◽

First Order

AbstractWe introduce a class of first order G-structures, each of which has an underlying almost conformally symplectic structure. There is one such structure for each real simple Lie algebra which is not of type {C_{n}} and admits a contact grading. We show that a structure of each of these types on a smooth manifold M determines a canonical compatible linear connection on the tangent bundle {\mathrm{TM}}. This connection is characterized by a normalization condition on its torsion. The algebraic background for this result is proved using Kostant’s theorem on Lie algebra cohomology. For each type, we give an explicit description of both the geometric structure and the normalization condition. In particular, the torsion of the canonical connection naturally splits into two components, one of which is exactly the obstruction to the underlying structure being conformally symplectic. This article is the first in a series aiming at a construction of differential complexes naturally associated to these geometric structures.

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Galois Lines for Space Curves

Algebra Colloquium ◽

10.1142/s100538670600040x ◽

2006 ◽

Vol 13 (03) ◽

pp. 455-469 ◽

Cited By ~ 4

Author(s):

Hisao Yoshihara

Keyword(s):

Galois Group ◽

Algebraic Structure ◽

Geometric Structure ◽

Space Curves ◽

Special Cases ◽

Three Space

Let C be a curve, and l, l0 be lines in the projective three space ℙ3. Consider a projection πl: ℙ3 ⋯ → l0 with center l, where l ∩ l0 = ∅. Restricting πl to C, we get a morphism πl|C: C → l0 and an extension of fields (πl|C)* : k(l0) ↪ k(C). We study the algebraic structure of the extension and the geometric structure of C. In particular, we study the structure of the Galois group and the number of Galois lines for some special cases.

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