The standard parametrization of torus knots lead to simple Björling surfaces only when the chosen normal is the surface normal of the underlying torus. But one can project a torus knot into a plane and lift it back to a closed space curve.
This turns out to be a differently parametrized torus knot, now suitable for all things Björling. Below are two Möbius strips for the (2,5) and (3,5) torus knots.
The possibility to let the normal rotate with higher speed leads to an interesting phenomenon. The “helicity” of the Björling surface seems to depend on how much torsion the space curve has.