Plane Curves with Consecutive Double Points

1914 ◽  
Vol 16 (1/4) ◽  
pp. 15 ◽  
Author(s):  
F. R. Sharpe ◽  
C. F. Craig
Keyword(s):  
Plane Curves ◽  
Double Points

2. Applications of the Theorem that Two Closed Plane Curves intersect an even number of times

1878 ◽  
Vol 9 ◽  
pp. 237-246 ◽  
Author(s):  
Tait
Keyword(s):  
Special Class ◽  
Double Point ◽  
Closed Curve ◽  
British Association ◽  
Plane Curves ◽  
Double Points ◽  
The Wire ◽  
Closed Plane

The theorem itself may be considered obvious, and is easily applied, as I showed at the late meeting of the British Association, to prove that in passing from any one double point of a plane closed curve continuously along the curve to the same point again, an even number of intersections must be passed through. Hence, if we suppose the curve to be constructed of cord or wire, and restrict the crossings to double points, we may arrange them throughout so that, in following the wire continuously, it goes alternately over and under each branch it meets. When this is done it is obviously as completely knotted as its scheme (defined below) will admit of, and except in a special class of cases cannot have the number of crossings reduced by any possible deformation.

Extremal Values of the Basic Invariants of Plane Curves

2000 ◽  
Vol 09 (08) ◽  
pp. 1085-1126
Author(s):  
Jianming Yu ◽  
Jianyi Zhou ◽  
Jianzhong Pan
Keyword(s):  
Fixed Number ◽  
Plane Curves ◽  
Explicit Formulas ◽  
Double Points ◽  
Fine Structures ◽  
Extremal Values ◽  
Extremal Value

In [A2] V.I. Arnold introduced three basic invariants St, J+ and J- of plane curves and proposed some interesting conjectures concerning the extremal value of these invariants on a given set of curves. Partial answers have been obtained by O. Viro and A. N. Shumakovich. We give explicit formulas for these extremal values of sets of plane curves with fixed number of double points and of Whitney index and we determine on which curves these extremal values are attained (Theorems 3-6). Our arguments are based on understanding of the fine structures of generic curves and some surgery operations on curves.

Parity conditions for realizability of Gauss diagrams

2019 ◽  
Vol 28 (01) ◽  
pp. 1950015
Author(s):  
Oleg N. Biryukov
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Plane Curves ◽  
Double Points ◽  
Gauss Diagram ◽  
Necessary And Sufficient ◽  
Closed Plane

We consider a problem of realizability of Gauss diagrams by closed plane curves where the plane curves have only double points of transversal self-intersection. We formulate the necessary and sufficient conditions for realizability. These conditions are based only on the parity of double and triple intersections of the chords in the Gauss diagram.

Unfolding plane curves with cusps and nodes

2015 ◽  
Vol 145 (1) ◽  
pp. 161-174
Author(s):  
Juan J. Nuño Ballesteros
Keyword(s):  
Finite Codimension ◽  
Plane Curves ◽  
Singular Set ◽  
One Dimensional ◽  
Double Points ◽  
Euler Obstruction ◽  
Type Singularity ◽  
Transverse Slice

Given an irreducible surface germ (X, 0) ⊂ (ℂ3, 0) with a one-dimensional singular set Σ, we denote by δ1 (X, 0) the delta invariant of a transverse slice. We show that δ1 (X, 0) ≥ m0 (Σ, 0), with equality if and only if (X, 0) admits a corank 1 parametrization f :(ℂ2, 0) → (ℂ3, 0) whose only singularities outside the origin are transverse double points and semi-cubic cuspidal edges. We then use the local Euler obstruction Eu(X, 0) in order to characterize those surfaces that have finite codimension with respect to -equivalence or as a frontal-type singularity.

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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