Are there "principal" bundles $S^1 \to S^3 \to S^2$ other then Hopf's? (They would be necessarily not locally trivial)

Question

It is well known that the only principal locally trivial fiber bundle $S^1 \to S^3 \to S^2$ is Hopf map $h$ (see, for example, [1]).

What if we drop the local triviality but mantain a "principality" hypothesis meaning that the fibers are given by an $S^1$ action? Are there well known families of such bundles $S^1 \to S^3 \to S^2$ whose homotopy class is not that of $h$?

In particular, are the homotopy class of the nontrivial multiples $k[h]$ of Hopf's map in $\pi_3(S^2)$, $k \not\in \{ \pm 1,0 \}$, given by such bundles?

Such bundles would be necessarily not locally trivial, but hopefully they can be fibrations.

Answer 1

The 3-sphere has infinitely many Seifert fibrations with generic fiber a torus knot (including the unknot).

For a $(p,q)$ torus knot, the Hopf invariant will be $pq$ (up to sign). To see this, note that the fibration has two exceptional fibers which form the Hopf link (these are the red line and yellow circle in the image). The generic fibers wrap $p$ times around one of these, and $q$ times around the other, so have linking number $pq$.

In fact, then, one can represent each homotopy class in $\pi_3(S^2)$ by a Seifert fibration with a single exceptional fiber (so all fibers are unknots, generically a $(p,1)$ curve on the Clifford torus).

These are (non-locally trivial) ``principal bundles" in the sense that there is an action of $S^1$ whose orbits are the fibers of the fibration. If $S^3\subset \mathbb{C}^2$ as the unit sphere, then the action is $(z_1,z_2)\to (z^pz_1,z^qz_2)$, for $z\in S^1= \{z\in \mathbb{C}, |z|=1\}$.