= Torus knot
{title2=$T_{p,q}$}
{wiki}
For nonzero <coprime integers> $p,q$, the torus knot $T_{p,q}$ lies on a standard unknotted <torus> in the <three-sphere> and winds $p$ times meridionally and $q$ times longitudinally. Here a <meridian of a solid torus> goes around the small cross-section of the doughnut, while the longitudinal direction goes around its central hole. The integers count signed turns, so changing one sign gives the <mirror of a link> of the <torus knot>. In this article $p$ is always the meridional count and $q$ the longitudinal count.
An explicit <parametric curve> in three-dimensional <Euclidean space> is
$$
\gamma_{p,q}(t)=\bigl((R+r\cos(pt))\cos(qt),\ (R+r\cos(pt))\sin(qt),\ r\sin(pt)\bigr),\qquad 0\leq t<2\pi,
$$
where $R>r>0$. Its cylindrical radius $\rho=\sqrt{x^2+y^2}$ satisfies $(\rho-R)^2+z^2=r^2$, the equation of the supporting <torus>. Its meridional angle is $pt$ and its longitudinal angle is $qt$: this checks the convention directly rather than relying on the picture. Its squared <speed> is
$$
\lVert\gamma'_{p,q}(t)\rVert^2=r^2p^2+q^2(R+r\cos(pt))^2>0.
$$
If two parameter values give the same point, their difference $\delta$ satisfies $p\delta,q\delta\in2\pi\mathbb Z$. <Bézout's identity> for the <coprime integers> $p,q$ then gives $\delta\in2\pi\mathbb Z$. Thus the loop closes and is an <injective function> on the parameter <circle>, with nonzero <derivative>; it is a <smooth embedding>.
For example, $R=2$, $r=0.7$, $p=2$, $q=3$ produces a <trefoil knot>. During one circuit the meridional angle changes by $4\pi$ and the longitudinal angle by $6\pi$. The supporting <torus> is a guide for construction, not a constraint on later <ambient isotopies>: the <knot> may move off it. This family makes arithmetic visible as geometry: the <coprime integers> determine whether a single traversal visits the whole loop without repetition.
\Image[/torus-trefoil-geometry.png]
{title=The trefoil T(2,3): two meridional turns and three longitudinal turns}
The <coprime integers> hypothesis matters. If the <greatest common divisor> is $d=\gcd(|p|,|q|)>1$, the same <parametric curve> retraces a single component $d$ times; it does not automatically draw $d$ different components. The corresponding <torus link> is instead the full set of points on the <torus> whose angles satisfy $q\phi-p\theta\in2\pi\mathbb Z$, with meridional angle $\phi$ and longitudinal angle $\theta$. It has $d$ disjoint components, each with winding counts $(p/d,q/d)$. For instance, the <torus link> $T_{2,2}$ consists of two parallel $(1,1)$ loops and is a <Hopf link>, whereas substituting $(2,2)$ into $\gamma_{p,q}$ simply traces one of those loops twice.
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