Hopf fibration of 3-sphere

[Hornbein]

Can you explain why they can always be separated in the "ambient" $\mathbb R^4$ ?

Sure. Locally the two circles are one dimensional. When they approach closely enough they look like lines. Suppose one line W is [w, 0, 0, 0] while the other line X is [0,x,0,e]. (w and x are free variables, e is a constant.) If the X line tries to move directly to [0,x,0,-e] then the two lines will intersect at the origin. But if the X line uses the 4th dimension by slipping over to [0,x,e,e] then it can move to [0,x,e,-e] then slip back to [0,x,0,-e]. So it isn't possible to link two circles in R4. It is possible to link two 2-spheres. In N dimensions one may link (N-2)-spheres. In two dimensions one may link two 0-spheres.

Great circles are always linked in S3 because they lack the freedom to maneuver like this. If either circle is not a great circle then they cannot be linked. Or at least so I believe.

 

[lavinia]

Sure. Locally the two circles are one dimensional. When they approach closely enough they look like lines. Suppose one line W is [w, 0, 0, 0] while the other line X is [0,x,0,e]. If the X lines tries to move to [0,x,0,-e] then the two lines will intersect at the origin. (w and x are free variables, e is a constant.) But if the X line uses the 4th dimension by slipping over to [0,x,e,e] then it can move to [0,x,e,-e] then slip back to [0,x,0,-e]. So it isn't possible to link two circles in R4. It is possible to link two 2-spheres. In N dimensions one may link (N-2)-spheres. In two dimensions one may link two 0-spheres.

Great circles are always linked in S3 because they lack the freedom to maneuver like this. If either circle is not a great circle then they cannot be linked. Or at least so I believe.

I think not all great circles. Some of them intersect with each other. The great circles in the Hopf fibration do not intersect with each other, this because they are orbits of the action of SO(2) on S^3. If two orbits intersected then where would the intersection go under the action of SO(2)?

 

[Hornbein]

I think not all great circles. Some of them intersect with each other. The great circles in the Hopf fibration do not intersect with each other, this because they are orbits of the action of SO(2) on S^3. If two orbits intersected then where would the intersection go under the action of SO(2)?

That's true. If great circles intersect then they are not linked.

Two great circles chosen at random will be linked with probability one. They intersect with probability zero.