spacetime’s knottiness

[AntonioLao]

The 1st book ever published about the theory of knots was in 1847 by Listing, a student of Gauss and ever since, many mathematicians are intrigued, specially, topologists. They have discovered many types of knots. But like the theory of curves and the theory of manifolds (surfaces), there exist topologically equivalent properties. These properties can help to distinguish and to classify the various types of knots.

The mathematical knots are different from the physical knots. But they share a common feature of solely being capable of analysis in no less than 3 dimensions. One single forbidden act is undergoing transformations by cutting or tearing. Mathematical knots are all closed loops, they have no end points. And since they are not allowed to be cutted, their being closed loops serve as a property of invariance or conservation (for all time) and by Noether’s theorem there is an underlying symmetry principle at work.

http://math.ucr.edu/home/baez/noether.html

Because of these unique properties, knots are suitable mathematical objects for the analysis of spacetime’s structures. And the simplest of all knots are the null knots. When 2 null knots are linked together with an attached directional property, two distinct topological structures can be created. These fundamental forms serve as the basic geometrical units for all spacetime’s scaffoldings.

[Guille]

What is a null knot?

[AntonioLao]

What is a null knot?

the curve of a circle or an ellipse is equivalent to a null knot.

[Guille]

the curve of a circle or an ellipse is equivalent to a null knot.

Doe sthis ellipse have to do with the eliptical equations?

[AntonioLao]

Doe sthis ellipse have to do with the eliptical equations?

the circle and ellipse share a general form of the equation of conic section, only the discriminants are different.

[Guille]

the circle and ellipse share a general form of the equation of conic section, only the discriminants are different.

what is this shared general form of the equation of conic section?

[AntonioLao]

GUILLE,

I'll post this general equation later. Have to look it up in the book.

[AntonioLao]

GUILLE,

The general equation of conic sections is the quadratic equation of two variables x, and y given as
[math]Ax^2+Bxy+Cy^2+Dx+Ey+F=0[/math]

Invariance functions of the coefficients A, B, C, D, E, F exist. The function [math]4AC-B^2[/math] is known as the characteristic. It identifies the type of conic sections. The discriminant [math]\\Delta[/math] determines their degeneracy.

[math]\Delta=\left|\begin{array}{ccc}A&B/2&D/2 \\ B/2&C&E/2 \\ D/2&E/2&F\end{array}\right|[/math]

[Guille]

GUILLE,

The general equation of conic sections is the quadratic equation of two variables x, and y given as
[math]Ax^2+Bxy+Cy^2+Dx+Ey+F=0[/math]

Invariance functions of the coefficients A, B, C, D, E, F exist. The function [math]4AC-B^2[/math] is known as the characteristic. It identifies the type of conic sections. The discriminant [math]\\\\Delta[/math] determines their degeneracy.

[math]\\Delta=\\left|\\begin{array}{ccc}A&B/2&D/2 \\\\ B/2&C&E/2 \\\\ D/2&E/2&F\\end{array}\\right|[/math]

Thanks for the info. What other invariant functions are there for the coefficients?

[AntonioLao]

What other invariant functions are there for the coefficients?

The 3rd invariant function mentioned in the book I'm using is [math]A+C[/math]. I'm sure there are many more. However, I have no knowledge what are they.