I've done the following construction. I expected to get the Clifford torus or another "Hopf torus", such as the ones we can see here: lobed Hopf tori.

Here is my construction. First I take the "inverse" Hopf map: $$ H^{-1}(q,t) = \frac{1}{\sqrt{2(1+q_1)}} \begin{pmatrix} -(1+q_1)\sin t \\ (1+q_1) \cos t \\ q_2 \cos t + q_3 \sin t \\ -q_2 \sin t + q_3 \cos t \end{pmatrix}, q \in S^2, t \in \mathbb{R}. $$

I generate all the great circles of $S^2$: $$ C_{\theta,\phi} = \begin{pmatrix} \cos \theta \sin \phi \\ \sin \theta \sin \phi \\ \cos \phi \end{pmatrix}. $$

Then I calculate the circles of $S^3$ obtained by the inverse Hopf map as follows: $$ [0, 2\pi] \ni t \mapsto H^{-1}(C_{\theta, \phi}, t) $$

Finally I apply the stereographic projection to these circles in $\mathbb{R}^4$ in order to map them to $\mathbb{R}^3$. And this is the picture I obtain:

enter image description here

Here is the same object but drawn in a different way:

enter image description here

Its not ugly but this does not look like a torus. And I've never seen such a picture elsewhere. What is it?

More importantly, I would like to know how to obtain the lobed Hopf tori (lobed Hopf tori). Is the way to get them close to my construction of am I totally in a wrong way ? In case I'm totally in a wrong way I would appreciate a couple of hints.

  • $\begingroup$ See my blog for an answer : 1/3 2/3 3/3. I think there was an error in my $H^{-1}$ formula. $\endgroup$ – Stéphane Laurent May 4 '18 at 10:55

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