Topological Invariants of Plane Curves and Caustics

1994 ◽  
Author(s):  
V. Arnold
Keyword(s):  
Plane Curves ◽  
Topological Invariants

FINITE ORDER TOPOLOGICAL INVARIANTS OF PLANE CURVES

1999 ◽  
Vol 08 (01) ◽  
pp. 33-47
Author(s):  
TETSUYA OZAWA
Keyword(s):  
Finite Order ◽  
Plane Curves ◽  
Topological Invariants ◽  
Mutually Independent ◽  
Closed Plane

We introduce three families of topological invariants of stable closed plane curves, which contain infinitely many mutually independent invariants among them. We study the order of these invariants in the sense of Vassiliev. As a consequence, we conclude that there exist infinitely many independent topological invariants for stable closed plane curves with order equal to 1.

A GENERALIZATION OF ARNOLD'S STRANGENESS INVARIANT

1999 ◽  
Vol 08 (05) ◽  
pp. 551-567
Author(s):  
HIDEYO ARAKAWA ◽  
TETSUYA OZAWA
Keyword(s):  
Triple Point ◽  
Infinite Sequence ◽  
Natural Extension ◽  
Plane Curves ◽  
Topological Invariants ◽  
Cusp Point ◽  
Tangent Point ◽  
Mutually Independent ◽  
Point Triple ◽  
Closed Plane

The purpose of this paper is to introduce an infinite sequence {Stk}k of mutually independent topological invariants of smooth closed plane curves, which is proved to be a natural extension of the rotation number and the strangeness invariant defined by Arnold in [Ar]. We prove a formula to express Stk by using the invariants [Formula: see text] which are defined in [Oz]. The jumps of Stk at perestroikas of three types (namely cusp point, triple point, and self tangent point perestroika) are investigated, and as a consequence we find that Stk have the order in the sense of Vassiliev equal to 1.

Base polynomials for degree and distance based topological invariants of n-bilinear straight pentachain

2021 ◽  
pp. 1-17
Author(s):  
Abdul Rauf Nizami ◽  
Khurram Shabbir ◽  
Muhammad Shoaib Sardar ◽  
Muhammad Qasim ◽  
Murat Cancan ◽  
...  
Keyword(s):  
Topological Invariants

scholarly journals Conjectures on Stably Newton Degenerate Singularities

2021 ◽  
Author(s):  
Jan Stevens
Keyword(s):  
Characteristic Zero ◽  
Arbitrary Number ◽  
Milnor Number ◽  
Plane Curves ◽  
Newton Diagram ◽  
Finite Characteristic ◽  
Vanishing Cycles

AbstractWe discuss a problem of Arnold, whether every function is stably equivalent to one which is non-degenerate for its Newton diagram. We argue that the answer is negative. We describe a method to make functions non-degenerate after stabilisation and give examples of singularities where this method does not work. We conjecture that they are in fact stably degenerate, that is not stably equivalent to non-degenerate functions.We review the various non-degeneracy concepts in the literature. For finite characteristic, we conjecture that there are no wild vanishing cycles for non-degenerate singularities. This implies that the simplest example of singularities with finite Milnor number, $$x^p+x^q$$ x p + x q in characteristic p, is not stably equivalent to a non-degenerate function. We argue that irreducible plane curves with an arbitrary number of Puiseux pairs (in characteristic zero) are stably non-degenerate. As the stabilisation involves many variables, it becomes very difficult to determine the Newton diagram in general, but the form of the equations indicates that the defining functions are non-degenerate.

On an area-preserving inverse curvature flow of convex closed plane curves

2021 ◽  
Vol 280 (8) ◽  
pp. 108931
Author(s):  
Laiyuan Gao ◽  
Shengliang Pan ◽  
Dong-Ho Tsai
Keyword(s):  
Curvature Flow ◽  
Plane Curves ◽  
Area Preserving ◽  
Closed Plane
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