In working with cycloids of different Kleinbottle types and my multiaxial
coordinates I have come to realize that the figure I called a tritorus was
actually a torus with three twisted axes! It is the next level beyond the
Kleinbottle! The realization that it is a cycloid of the tritorus that is my
version of the trefoil showed me that what you are looking for may be right in
front of you face if you are not too stubborn to see it!
The tritorus is defined like this:
1) f(t) = cos(t)
2) g(t) = cos(t + 2 * Pi / 3)
3) h(t) = cos(t + 4 * Pi / 3)
4) f1(t) = sin(t)
5) g1(t) = sin(t + 2 * Pi / 3)
6) h1(t) = sin(t + 4 * Pi / 3)
7) x = (1 + f1(t)) *f(t)
8) y = (1 + g1(t)) * g(t)
9) z = (1 + h1(t)) * h(t)
The resulting surface even after seeing
three or more views is hard to visualize!
The trefoil cycloid is:
10) fc(t)=((n1) * f((n1) * t) + (n + 1) * f((n+1) * t)) / (2 * n)
That this gives a more angular trefoil than other definitions is due to the use
of the triaxial functions. The use of three of the five pentaxial functions in
the tritorus functional form gives a 3D projection of a pentafoil, but it "hits"
in the center unlike the ideal of a knot, but very much like what we see in
Kleinbottle cycloids. A five twisting manifold or pentatorus can be defined as
I defined the pentafoil, but using just the pentaxial functions and not their
cycloids! A point that is important is that since the "tubes" definitions are so
slow and I was so eager to get the multifoil work done, I didn't do any of the
other cycloidal n values for either of these!
The question of how a field can be constructed of these trajectories is a
good one. A Russian in Nature has published such a "theory". I tried to get the
Curl and Div operators in Mathematica to work on my functional definitions with
no luck! The t variable is the theta angular variable in ordinary spherical
coordinates: to give these functions a dynamic/ physical reality as surfaces and
volumes more variables in a partition function are necessary! Suppose we ignored
that and made up a system Hamiltonian energy equation on the vector of the
trefoil {x,y,z}:
11) K X{x,y,z) + V(t) * {x,y,z} = E * {x,y,z}
for a vector cross kinetic operator K and a
functional potential function V(t) to
give energy scalar E. This equation becomes three interdependent and relatively
symmetrical differential equations.Suppose that they are just dynamic kinetic
energies of independent trajectory directions: in each of three directions
12) (Mx / 2) * (dx / dt)^2 + V(t) * x = E * x
13) (1 / x) * dx^2 = 2 * (E  V(t)) * dt^2 / Mx
The problem of rest mass and potential function remain for these three simple
equations! It looks like for constant E(n) and Mx the cycloids determine a
potential function as x is a function of the angular time!
14) V(t) = (E(n)  (Mx / 2) * (dx / dt)^2)
14a) (dx / dt)^2 = (4 * (3 * Cos[2 * t] + 13 * Cos[4 * t] + 6 * Cos[6 * t] +
8 * Cos[8 * t])^2) / 81 for n = 3
The other squared velocities are 2 * Pi / 3 and 4 * Pi / 3 plus t functionally
and the squared path length
is about 2 * E(3) / Mx = 6.8 relative to E(1). It is the tritorus form and the
cycloid that make this function somewhat more
complex. The result is more like three independent oscillators that that of a
wave function, so must be considered an approximation!
Let us look at the assumptions we are making:
1) a three manifold that twists back on itself
2) captured trajectories of a quantum cycloidal nature
3) a mass that follows the trajectories
4) classical Hamiltonian dynamics
The resulting cycloidal vector is a wave function of a field
Essentially we are saying we have a "solution", so we can design a "system"
around them! We haven't specified the scale or a
quantum mechanical energy of the
Schroedinger equation sort! We have a reference level in the tritorus of the
triaxial functions. A simple model of a multiaxial twisted torus topology as a
dynamic system like what Lord Kelvin might have defined before Heissenberg.
Planck and Born! The symmetry that allows the individual dimensional separation
is due to my multiaxial coordinate invention and is not visible in other
definitions of the multifoil parametrics! A spectrum of energies is predicted by
these equation for a given mass scale. The theory that knot/ loop symmetry is
that of quantum gravity has been put forth: that assumption would make necessary
some sort of time dependent quantum energy instead of the fixed scalar
approximation. That the constant is
Planck's in this quantum energy may not hold,
if these are not photonic wave functions, but gravitational wave
functions of a
curvature of space! In theory a photon should be composed of the more
fundamental
string and not the other way around!
What I have tried to do in this article is to show a connection of knot
theory to twisted topology in toruses, to extend this to a quantum dynamic
approach in a very simple classical model,
and to examine the kind of analysis of
assumptions that is necessary to understanding of a gravitational string
application of these functions. The simplicity is possible only because of my
invention of the multiaxial coordinates and their
application to torus topologies.
