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I don't think this has anything to do with mathematics. A tribual bundle is a trivial bundle whether it is given as a product or it is...

It took me a while to understand what you are saying. Yes. You can't rotate one circle in the link so that it is flipped without banging...

The trouble I had with the Hopf linkage was I was using the wrong definition of linkage. I was looking for two distinct great circles...

True. Principal angles are magnitudes always in the range [0,pi/2], just like in 3D.

Technically if sin(a) and sin(b) are both not zero. Without invoking a probability space, the set of pairs of angles with one or both...

Can you make that probaility argument more precsise?

I think not all great circles. Some of them intersect with each other. The great circles in the Hopf fibration do not intersect with...

Sure. Locally the two circles are one dimensional. When they approach closely enough they look like lines. Suppose one line W is [w...

Here is an image of a twisted cylinder that shows the linked boundary circles. This was taken from Wikipedia.

Clifford parallels and flat tori in the three sphere Start with a square in the xy-plane whose corners are (0,0),(2π,0),(0,2π) and...

They are not homeomorphic. The three dimensional space deforms continuously onto a 2 dimensional torus. The idea of continuous...

After continuous deformation it is a regular 2d torus. I had trouble describing the situation in a clear way even though the picture...

Another way to think about the linking of two fibers in the Hopf fibration is to look at the topology of their complement in S^3. Take...

yes I just wanted to emphasiize that by sliding the linked pair of fibers along the rays emanating from the north pole of the three...

Here is a way of doing it using stereographic projection. Slide the two linked fibers along the rays emananting from the north pole of...