Constructing the Fibration Surface

Now we see our inside and outside surfaces whose boundaries on the torus are exactly the boundary curves found previously.

When the torus is unwrapped as a square these surfaces look like curtains below and above the plane. By transforming the square to torus with the gluing operation as before, we see begin to see our fibration subsurfaces.

Note that the rungs from the previous visualization correspond exactly to where inside and outside subsurfaces meet, and thus is not a boundary of their union.

This visualization gives good intuition for how the complete fibration surface is constructed from the subsurfaces. However, it is flawed because there appears to be a circular boundary on the outside subsurfaces as they 'tear apart'. These circular boundaries must also be glued closed so that the only boundary for our complete fibration surface is the trefoil knot.