Glue Latitudinal Edge
Glue Longitudinal Edge
Knot Curve
Inside Boundary
Outside Boundary
Hide Rungs
Animate \(\theta\)

Boundaries of Fibration Surfaces

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 \(\theta \in S^1 \)