Cabling a knot’s surface

Cabling a knot isn’t so tricky to imagine. Let’s just consider a tubular neighborhood of the knot.

Glue the top to the bottom. On the left we have the knot, on the right we have a (3,1)-cable… using the straight vertical framing.
Here’s a (3,5)-cable.

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.

Let’s do this with the (3,1)-cable.

Here’s the knot with it’s Seifert surface hanging off. The surface continues off to the left, but we’re only looking near the knot.
Take a tubular neighborhood and restrict to looking outside it.

tubularnbhd one page

For the (3,1)-cable, we’ll take 3 pages and one meridional disk.

3pages withonemeridian

Then we’ll attach them with twisted bands so the resulting boundary can run along the cable.

attachwithbands takethecable

And there you go. But it’s kinda nice to smooth it out.

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.
Fibration by Seifert surfaces

With the cable smoothed, we can see how the surface corkscrews upwards to fill in the fibration.
Fibration of (3,1)-Cable

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.
Fibration of the (3,5)-cable.

You can get the SketchUp model here and take a closer look yourself.

~ by Ken Baker on November 18, 2009.

7 Responses to “Cabling a knot’s surface”

  1. Where was this when I was a grad student? Great stuff.

  2. That looks like your older JVH animations. Do you think that family of covers comes from cabling?

    • I used the same color scheme except with blue binding, so that helps it look similar.

      Off the cuff, I’d reckon that the various covers of the JVHM open book don’t come from cabling.

      But perhaps there are ways of constructing an open book on a manifold with a self cover so that the lifted open book could also have been obtained by cabling. Anyone know a nice example or a reason why not?

  3. Dear backer,
    i am not good in mathematics but i am interested in forms and i look for those. i am fasinated in topological geometries .I am intrigued by your fibration made concious of the space around knots…can you explain me how this fibration related to the geometry of the knot and whats its meaning in real world.if there any book or website references pls let me know.

    • Hi dk. I don’t think I’ll have a satifactory short answer for you… and it would hinge upon what you’re meaning by “the geometry of the knot”.

      But here’s one, albeit a bit technical:
      We’ve got these cables which are knots/links in the solid torus. The “geometry” is that their exteriors are Seifert fibered spaces over a (multiply) punctured disk with one exceptional fiber. The fibers are horizontal surfaces, meaning they’re transverse to the Seifert fibration.
      One may also observe that the fibration is periodic and the flow may be arranged so that the flowlines are closed loops. These flowlines give a Seifert fibration

      As for “meaning in real world”… it’s whatever you ascribe to it. All y’all else out there are welcome to chime in on this with yer two bits.

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