Knots and Twisted Manifolds

R. L. Bagula ©
23 July 1997

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)=((n-1) * f((n-1) * 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.