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The following text describes a construction of the unit sphere $\mathbb{S}^{3}$.

To consider the next example, it is helpful to practice visualizing $\mathbb{S}^{3}$ as consisting of two "polar" great circles $\{(z,0): z \in \mathbb{S}^{1} \subset \mathbb{C}\}$, $\{(0,w): w \in \mathbb{S}^{1} \subset \mathbb{C}\}$, together with a family of nested Clifford tori interpolating between them. Thinking of $\mathbb{S}^{3}$ as the unit sphere in $\mathbb{C}^{2}$, these Clifford tori are $$ T_{\alpha}:=\left\{(z \cos \alpha, w \sin \alpha): z, w \in \mathbb{S}^{1} \subset \mathbb{C}\right\} $$ for $\alpha \in(0, \pi / 2)$. (Note that $T_{\alpha}$ is the set of points at spherical distance $\alpha$ from the circle $\{(z, 0)\} .$ ) If we stereographically project to $\mathbb{R}^{3}$, then the two circles become the $z$ -axis and the unit circle in the $x y$ -plane, while the nested tori $T_{\alpha}$ grow around the circle and shrink around the line.

I’ve been trying but could not visualize how the two circles and the Clifford tori describe $\mathbb{S}^3$. Is there an equivalent construction for $\mathbb{S}^{2}$?

The picture provided with the text looks similar to the last one in this answer. However the process described there is also not clear to me. What is the relationship between $\mathbb{S}^{3}$ and two solid tori?

anon
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ensbana
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  • There's not really an analogous construction in $S^2$, unfortunately. How happy are you with the definition of Clifford tori? And how happy are you with dividing a Clifford torus into Hopf circles? – Andrew D. Hwang Apr 03 '21 at 17:14
  • I’m not good with either unfortunately. This is the first time I encounter these concepts. The text gives some further information about the construction, but how those pieces come together to form $\mathbb{S}^3$ is still unclear to me. – ensbana Apr 03 '21 at 18:00
  • The text is: ”The Hopflink $H$ is the link of two components represented by two polar great circles in $\mathbb{S}^{3},$ or equivalently by any two linked round circles in $\mathbb{R}^{3}$. Its complement is the union of all the Clifford tori between these two circles. That is, $\mathbb{S}^{3} \backslash H \cong \mathbb{R} \times T^{2} \simeq T^{2}$. It follows that $\pi_{1}\left(\mathbb{S}^{3} \backslash H\right) \cong \mathbb{Z}^{2}$”. – ensbana Apr 03 '21 at 18:00
  • Why don't you provide a source for the text you quote? – anon Apr 14 '21 at 02:18

1 Answers1

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Here's a capsule summary. Via stereographic projection, the three-sphere may be identified with $\mathbf{R}^{3}$ with one point at infinity. The left-hand figure shows a closed half-plane. The blue line at left is a circle (with one point at infinity). When this half-plane is revolved using the blue line as axis, each circle sweeps out a Clifford torus, and the blue spot where the circles acculumate sweeps out a circle that links the axis.

(Tangential fun facts: If $p = (1, 0)$ and $q = (-1, 0)$, then each circle is the solution set of $|x - q|/|x - p| = k$ for some real $k > 1$. The blue axis is the solution set for $k = 1$, and the point $p = (1, 0)$ is in a sense the limit as $k \to \infty$. These circles are orthogonal to the family of circles through $p$ and $q$.)

The tori partition the complement of the two circles in $S^{3}$ because the circles partition the open punctured right half-plane, and this partition is a product foliation (essentially by the fun facts above).

Now let's focus on one of the circles sweeping out a Clifford torus. The animation on the right shows the fibation of a typical Clifford torus into Hopf circles, which are images of great circles on $S^{3}$ under stereographic projection.

One remaining detail you may want to ponder is that as $k \to 1^{+}$, the Hopf circles' radii grow without bound and the planes they lie in approach longitudinal planes, while as $k \to \infty$ the Clifford torus collapses to the unit circle swept out by $p$.

Circles sweeping out Clifford tori Hopf circles fibring a Clifford torus

Andrew D. Hwang
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  • Sorry for the late reply. I’ve read some more, but the idea of gluing the boundary of two solid tori together still appeals to me more. Would this be a correct and equivalent representation of what’s going on? https://imgur.com/wqhdlDC I start with two tori $T^2_1$ and $T^2_2$ interlocking, then “cutting” $T^2_1$ open and try gluing each of its boundary circles to one in $T^2_2$. To retain its topological shape, the inner circles of $T^2_1$, after the transformation, would have to become bigger and bigger circles, with the inner most one becomes a line with a point at infinity. – ensbana Apr 05 '21 at 10:03
  • Also the fun fact in your answer unfortunately completely went over my head. Are we seeing the points $p$ and $q$ in $\mathbb{R}^4$? Could they somehow fit into my picture? – ensbana Apr 05 '21 at 10:05
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    Yes, those diagrams and your description are correct (and compatible): The solid tori here are the solid torus swept by the interior of any particular circle, and the solid swept by the exterior of that same circle, and the boundary gluing is effected by the fact of a shared boundary. IIRC, the points $p$ and $q$ are in the plane, right there as we see them (well, except $q$ is off-screen to the left). :) When revolved, each sweeps out the core circle of the "inside" torus. – Andrew D. Hwang Apr 05 '21 at 12:42
  • (a) I think showing Villarceau circles should wait until the end of an answer, after bringing up the Hopf fibration, since they and the Hopf fibration are more difficult to understand than what's strictly necessary for describing nested Clifford tori, and this might not be obvious to the reader. – anon Apr 14 '21 at 02:41
  • (b) For fixed $z$ and $w$ and as $\alpha$ varies, we get right-angle arcs, which illustrates $S^3$ is a join $S^1\ast S^1$. Under stereographic projection, they are still arcs between points on the line and circle. If we take a 2D cross-section, then these arcs and the nested circles (sections of the tori) form so-called bipolar coordinates. The closest/furthest points on the nested circles, as well as their centers, are nice trigonometric functions of $\alpha$. These coordinates are also the stereographic projection of longitude/lattitide on the two-sphere! – anon Apr 14 '21 at 02:41