A primary requiremnt for our fibration surface is that its boundary is exactly the
torus knot. We will construct our family of fibres from this requirement by first considering the surface boundaries.

We construct our complete fibration surface by taking the union of subsurfaces, corresponding to the portion of the fibre
inside and
outside of the torus. We consider the boundaries of these subsurfaces, which will be
blue and
orange curves embedded in our torus.

Since our complete fibration is the union of the subsurfaces, wherever
inside and
outside boundary curves coincide is actually not a boundary to our complete fibration. Therefore, we can 'cancel' these portions of the boundary curves. We call these
rungs for their similarity to ladders.

If we consider the only the boundaries of the
inside and
outside surfaces without the
rungs, we see that the final boundary is exactly the
trefoil knot, as desired. This is true for all values of \(\theta \in S^1 \)