Cabling a knot’s surface
Cabling a knot isn’t so tricky to imagine. Let’s just consider a tubular neighborhood of the knot.
It’s not too bad to think about how a Seifert surface extends across the cable.
For a (p,q)-cable, we’ll take p parallel copies of the Seifert surface outside the tubular neighborhood and q copies of the meridional disk in the tubular neighborhood. The signs of p and q tell you the orientations you want on these pieces. Then you attach them together with |pq| twisted bands, twisted in the appropriate direction.
For the (3,1)-cable, we’ll take 3 pages and one meridional disk.
Then we’ll attach them with twisted bands so the resulting boundary can run along the cable.
Smoothing it out also has the benefit of helping you see how cabling extends the fibration of a fibered knot.
The way we’ve been looking a the knot, a fibration looks like this.
With the cable smoothed, we can see how the surface corkscrews upwards to fill in the fibration.
And we can even stack this before gluing the top to the bottom to get other (3,q)-cables. Here’s the fibration of the (3,5)-cable on its side.
You can get the SketchUp model here and take a closer look yourself.