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.