Torus Knots

Hello!

Hope all of you have been having a good week!

This week, We'll be talking about different kinds of knots, not necessarily as classified by an invariant, but known as a Torus Knot.

Before this, though, we should probably explain what a torus is.

The torus, pictured below, is the mathematical equivalent of the outside shell of a donut or bagel, whichever you prefer.

Figure 1 - The Torus

Therefore, a torus knot is a knot that has the knot itself (string), laying on the surface of the torus shape, without crossing over or under itself (i.e. - intersecting itself in any place).

The torus is, incidentally, made up of two types of circles, the meridian curve, which wraps once the short way around the torus, and the longitude curve, which wraps once the long way around the torus.

Torus knots are named as T(p,q), where the knot crosses over the meridian curve (the purple circle in Figure 2) p times, and the longitude curve (the red circle in Figure 2) q times.

Figure 2 - The longitude and meridian curves of the torus. 

An example, would be the (3,5) Torus Knot below in Figure 3. It crosses around the meridian curve five times, while crossing around the longitudinal curve 3 times, and is denoted as the (3,5) torus knot,

Figure 3 - A T(3,5) Knot, or a (3,5) Torus Knot

It's easy to draw a torus knot. For example, for the torus knot (p,q) = (5,3), first, take the torus, and first draw p points, in this case 5 points, on both the inside and the outside meridian curves of the torus, as seen below in Figure 4, in corresponding positions.

Figure 4 - Draw p points on the inside and the outside.

Note how there are two sides of the torus, like the top and bottom has of a hypothetically halved and emptied bagel. To create the Torus Knot, we attach these inside and outside points in different ways.
On the bottom side of the torus attach each of the 5 inside points to each of the 5 outside corresponding points, as shown below in Figure 5.

Figure 5 - On the top half, attach each outside point to the corresponding inside point.

Now, on the top side of the torus, take each outside point, and attach it to the inside point that is a (3/5) clockwise turn, as shown below in Figure 6.

Figure 6 - On the bottom half, attach each outside point to the point that is rotated 3/5th clockwise.

And, now, you've successfully drawn a (5,3) torus knot!

Incidentally, though, we can also show that the knot T(p.q) can be deformed (moved around) to be identical to the knot T(q,p). Essentially, we now know that the T(p,q) and T(q,p) knots are equivalent.

In the diagram below, for example, we see a (3,2) torus knot being transformed into a (2,3) torus knot. It's pretty cool, actually, and makes our job a lot easier - we don't have to keep track of which order to put which in.

Figure 6 - The T(3,2)  Torus Knot is a (2,3)  Torus Knot  

Besides being cool and fun to draw, torus knots important for being relatively simple to classify and look at, and, more specifically, give us a "family" of knots that we can consider at one time. In the papers I'm reading right now, especially, certain statements can be made for torus knots, given their properties, and these specific statements can lead to larger, more interesting generalizations! 

Thanks for reading this week, and hope you keep up for next week's post!