3-sphere complex co-ordinates I am currently trying to understand some mathematical physics papers that deal with torus knots.
I am trying to find the origin of a complex scalar field used. These fields are somehow related to the Hopf fibration.
I have spent the last week reading about and trying to understand the Hopf fibration. I Believe I understand the Hopf fibration with the mapping of $h(z_1,z_2)\mapsto \frac{z_2}{z_1}$.
Multiple papers are stating this:
$$u = u(r) = \frac{(r^2 −1)+2iz)}{r^2 +1},\ \ \   v = v(r) = \frac{2(x +iy)}{r^2 +1}$$
As the "standard complex co-ordinates for the 3-sphere". I am unable to find this information anywhere else apart from these papers, and I need to find out where it arises from, as to me this doesn't make sense. 
I am a 2nd year physics undergraduate and if you could explain in a way I can understand, that will be awesome!
Here is one paper - the rest are behind paywalls https://arxiv.org/pdf/1302.0342.pdf Eq (10)
 A: The following answer is not rigorous. To make it rigorous you may need to know more about smooth manifold. 
$\mathbb S^3 = \{ (u, v) \in \mathbb C^2 : |u|^2 + |v|^2  =1\}$ is itself a $3$-manifold, which means that locally it looks like an open sets in $\mathbb R^3$. Mathematically speaking, it means that for each point $p\in \mathbb S^3$, there is a parametrization $ \phi : U \to \mathbb S^3$ (where $U$ is open in $\mathbb R^3$), which is at least an injective mapping (with some other condition), so that $p\in \phi (U)$. This is called a (local) chart on the manifold, as you are giving for each points in $\phi (U)$ a coordinate $(x, y, z)$. 
Of course this three numbers $(x, y, z)$, which is associated with the point $\phi(x, y, z)$ in $\mathbb S^3$, depends on the chart $\phi$. But $\mathbb S^3$ is so explicit that it has a "standard" chart, given by stereographic projection. 
The stereographic projection is a mapping $(x, y, z)\mapsto (u, v) \in \mathbb S^3$ given by your formula, where $(x, y, z) \in U :=\mathbb R^3$ and $r = \sqrt{x^2 + y^2 + z^2}$. 
One needs to check that the above mapping is really a chart (that is, it is injective). One can use the following observation about the construction of stereographic projection: given $(x, y, z) \in \mathbb R^3$, the point $(u, v)$ in $\mathbb S^3$ is formed by the intersection of $\mathbb S^3$ with the line joining $(x, y, z, 0)$ to $(0,0,0,1)$. See the picture here for the two dimensional picture. 
