The simplest way to parametrize a (p,q) torus knot is to parametrize a torus as a surface of revolution, and to draw the image of a line with rational slope p/q.

Above is a torus cut open along a (2,5) knot, and to the right the Björling surface one gets from it, using as a normal the surface normal to the torus. Below are (2,3) and (3,5) knots, obtained the same way.

All these surfaces are complete and have finite total curvature. A drawback of this parametrization is that there appears to be no simple way to make the normal spin to obtain also twisted minimal surfaces of finite total curvature.

Resources

Mathematica Notebook

PoVRay Sources