Geometrical Representation
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 and . This lies on the surface of the torus given by (in cylindrical coordinates).
Other parametrizations are also possible, because knots are defined up to continuous deformation. The illustrations for the (2,3)- and (3,8)-torus knots can be obtained by taking, and in the case of the (2,3)-torus knot by furthermore subtracting respectively and from the above parametrizations of x and y. The latter generalizes smoothly to any coprime p,q satisfying .
Read more about this topic: Torus Knot