Knotception (Alternatively, Satellite Knots)

Hello!

Thanks to everyone for keeping up with my knot blog for the past few weeks. 

Last week, we talked about Torus Knots (click  for a quick review! ) and how to create them. This week, we'll be talking about something similar, Satellite Knots. They involve knots inside of other knots (knot sure what that means? read on!), which brings us to the title of Knotception!

Let us consider the Torus shape we talked about last week, and make this a solid (i.e. - An actual donut with everything inside instead of the hollow shell we had before).

We can put a knot K1 inside this torus, so that it is wrapped around the "hole" in the center, see Figure 1. 

Figure 1 - Solid Torus with Knot K1 inside

Now, we can, and this is the fun part, take the torus shell we have, and twist this into the shape of another knot K2, a torus, to be specific. It looks kind of like this:

Figure 2 - A torus twisted into a trefoil.

Except that we still have to remember the knot we had inside the torus. So, it ends up looking something like this:

Figure 3 - The first satellite knot that we have created.

With the 2-dimensional knot inside of the torus. The whole thing, or knot K3, is called a satellite knot, and the knot K2 is called the companion knot of the satellite.

You can think of it, maybe, as knot where the the satellite knot, or the final knot, orbits around (or is inside) the companion torus knot that is the center of the knot as a whole.

Alternatively, we could have an unknot inside the original torus knot. However, though, we have the knot inside twisted around a few times so that it looks as below, and we have the knot and the solid torus together as knot K1.

Figure 4 - The torus, with a knot twisted around inside 

Here, though, if we twist this again into the torus shape as we did before, we have a knot that looks like this:

Figure 5- The Whitehead Knot

We call this a Whitehead Double of a Trefoil (the girl-scout cookie knot shape) because of the original knot's resemblance to the Whitehead link, comprised of two knots and shown below.

Figure 6 - The Whitehead Link. Kind of resembles the knot above.

Now, you may think, where can we go from here.

Well, it turns out, it does matter how we squish the unknot in the middle.

Because if we take the knot we had from before, and, this time, before we twist the torus (donut) into a trefoil (girl scout cookie-shaped knot), we cut it in the middle around the meridian circle (circled in red below), and twist each side around in the direction of the arrows (also shown below) in red.

Figure 7 - Take the knot from before, cut it open across a vertical circle, and twist it around a few times.

We then, take this and twist it, again, into a torus shape, and get the following result.
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Figure 8 - This is the second Whitehead Double of the Trefoil.

This is a second Whitehead double of the trefoil. Resultingly, the knot inside is called a two-strand cable of the companion knot.

And, on that note, we'll end for this week. Thanks for keeping up, and hope you keep reading for more!