Trefoil knot

Example

The trefoil knot is the image of the map $f \colon \R / 2\pi\Z \to \R^3$ given by $$f(t) = \bigl((2 + \cos 3t) \cos 2t, \ (2 + \cos 3t) \sin 2t, \ \sin 3t\bigr).$$ The map is continuous, and it is injective. Indeed, if $f(t) = f(s)$, the last coordinate and the distance $2 + \cos 3t$ from the $z$-axis give $3t \equiv 3s \pmod{2\pi}$. The angle around the $z$-axis gives $2t \equiv 2s \pmod{2\pi}$. Then $t - s$ is a multiple of $2\pi/3$ and of $\pi$, so of $2\pi$. Hence the image is a knot. It lies on a torus, and it winds $2$ times around the axis of the torus and $3$ times around its core. So it is the torus knot of type $(2, 3)$.