Torus knot

In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R³. Similarly, a torus link is a link which lies on the surface of a torus in the same way. Each torus knot is specified by a pair of coprime integers p and q. A torus link arises if p and q are not coprime (in which case the number of components is gcd(p, q)). A torus knot is trivial (equivalent to the unknot) if and only if either p or q is equal to 1 or −1. The simplest nontrivial example is the (2,3)-torus knot, also known as the trefoil knot.

A torus knot can be rendered geometrically in multiple ways which are topologically equivalent (see Properties below) but geometrically distinct. The convention used in this article and its figures is the following.

The (p,q)-torus knot winds q times around a circle in the interior of the torus, and p times around its axis of rotational symmetry. If p and q are not relatively prime, then we have a torus link with more than one component.

The direction in which the strands of the knot wrap around the torus is also subject to differing conventions. The most common is to have the strands form a right-handed screw for p q > 0.

The (p,q)-torus knot can be given by the parametrization

where ${\displaystyle r=\cos(q\phi )+2}$ and ${\displaystyle 0<\phi <2\pi }$. This lies on the surface of the torus given by ${\displaystyle (r-2)^{2}+z^{2}=1}$ (in cylindrical coordinates).

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