Newtonian knot

The Newtonian knot in the real affine plane

The Newtonian knot (named after Isaac Newton ) is a plane algebraic curve of degree three, i.e. one cubic . It is rational , i.e. birational equivalent to the projective straight line . With Neil's parabola (except for coordinate transformation) it is the only rational cube in the plane.

definition

The Newtonian knot is an algebraic curve in two-dimensional affine or projective space. It is given by the equation

${\displaystyle y^{2}=x^{2}(x+1)}$

described, in homogeneous coordinates:

${\displaystyle zy^{2}=x^{3}+zx^{2}}$

properties

According to Newton's classification of cubic curves, the Newtonian knot belongs to the diverging parabolas. ${\displaystyle C}$

rationality

It has a rational parameterization

${\displaystyle \phi \colon \mathbb {P} ^{1}\to C\subset \mathbb {P} ^{2}}$

${\displaystyle \phi \colon (t_{0}:t_{1})\mapsto (t_{0}^{3}:t_{0}(t_{1}^{2}-t_{0}^{2}):t_{1}(t_{0}^{2}-t_{1}^{2}))}$

The parameterization shows that the Newtonian knot is rational , that is, birationally equivalent to . ${\displaystyle \mathbb {P} ^{1}}$

Dual curve

The dual curve has the parameterization: ${\displaystyle C^{*}}$

${\displaystyle \phi ^{*}\colon \mathbb {P} ^{1}\to C^{*}\subset \mathbb {P} ^{2}}$

${\displaystyle \phi ^{*}\colon (t_{0}:t_{1})\mapsto ((t_{0}^{2}-t_{1}^{2})^{2}:t_{0}^{2}(t_{0}^{2}-3t_{1}^{2}):-2t_{0}^{3}t_{1})}$

${\displaystyle C^{*}}$is a heart-shaped quartic and is called a cardioid .

Singularity

The singularity is a colon . The following applies to the above figure : ${\displaystyle \phi }$

${\displaystyle \phi (1:1)=\phi (1:-1)}$

In the vicinity of the singularity, the curve clearly looks like the intersection of two curves.

If one looks at the curve over the complex numbers, one can see that the root of holomorphic is for , so one can write: ${\displaystyle (x+1)}$ ${\displaystyle x<1}$

${\displaystyle y^{2}-x^{2}(x+1)=(y+x{\sqrt {x+1}})(y-x{\sqrt {x+1}})=f_{1}\cdot f_{2}}$

with two holomorphic functions and . ${\displaystyle f_{1}}$ ${\displaystyle f_{2}}$

Algebraically, this corresponds to the isomorphism of completed local rings .

literature

• Robin Hartshorne : Algebraic Geometry , Springer-Verlag, New York / Berlin / Heidelberg 1977, ISBN 3-540-90244-9
• Gerd Fischer: level algebraic curves , Vieweg (1994), ISBN 3-528-07267-9