Toroidal Coordinates on S^3
We’re looking at what amounts to being the genus 1 Heegaard sweep-out of S^3. There are a few concentric tori, two core curves, and two intersecting meridional disks. Of course above we’re looking at an odd perspective from the level of an orangish torus.
Pulling back, just looking at the two core curves and a couple of disks they bound, we see the following.
Since is the unit sphere in , we can think of these two disks as the unit –disk and the unit –disk. You might think of each of these in terms of polar coordinates: . Then the angles and are independent while the two radii are bound by . In some sense this gives what one might call toroidal coordinates on .
This is one framework for realizing the knots and curves that come from links of singularities (such as the trefoil coming from ) and visualizing knots from grid diagrams among others.
Just because I could, I made a quick fly through. (You might have to run it a second time if it races past you. Still trying to figure out how to loop.)
And here are some more stills. Yet more are on my Flickr.