I would like to understand the topological type of a link of a singularity in a simple example. Consider for instance the cone ${xy-z^2=0}\subset\mathbb{C}^3$. If we set $x = x_1+ix_2, y = y_1+iy_2, z = z_1+iz_2$ then this corresponds to the $4$-dimensional real subvariety of $\mathbb{R}^6$ given as the complete intersection $\{x_1y_1-x_2y_2-z_1^2+z_2^2 = x_1y_2+x_2y_1-2z_1z_2 = 0\}$.

Therefore the link of the singularity is given by

$\{x_1y_1-x_2y_2-z_1^2+z_2^2 = 0, x_1y_2+x_2y_1-2z_1z_2 = 0, x_1^2+x_2^2+y_1^2+y_2^2+z_1^2+z_2^2 = 1\}$

What is the topological type of this link?

Can we determine the topological type of the link from these three equations?