Hopf link

Example

Let $C_1 = \set{(x, y, 0) \in \R^3 : x^2 + y^2 = 1}$ and $C_2 = \set{(x, 0, z) \in \R^3 : (x - 1)^2 + z^2 = 1}$. Each of them is a round circle. They are disjoint: a common point would have $y = 0$ and $z = 0$, so $x = \pm 1$ from $C_1$ and $x \in \set{0, 2}$ from $C_2$, which is impossible. So $C_1 \cup C_2$ is a link with $2$ components, the Hopf link. The circle $C_2$ passes once through the flat disk bounded by $C_1$, at the point $(0, 0, 0)$, and it meets that disk nowhere else. Hence the linking number of the two components is $\pm 1$, the sign depending on the orientations. The unlink of two components has linking number $0$, so the Hopf link is not equivalent to it: the two rings cannot be pulled apart.