Hyperbolic Knots

Hi everyone! It's been a bit of time since my last post, but we'll be catching up on a lot.

In the last two weeks, we went over torus knots and satellite knots. As a brief review, these are, respectively, knots that are wrapped around a torus p times horizontally, q times vertically, and knots that are embedded into other knots. (For details, you can refer to the previous posts here and here.)

We mentioned prime and composite knots a while ago, and we said that composite knots were knots that could be created by composing two nontrivial knots, and prime knots were those that could not.

Interestingly enough, we find that all composite knots are satellite knots, while not the reverse is true. You might wonder: So what about prime knots?

That’s an excellent question, I’m glad you asked! Prime knots are more spread out, but, interestingly, most prime knots are hyperbolic knots. Which, incidentally, is our first topic today!

The formal definition of a hyperbolic knot is “a knot with a complement that can be given as a metric of constant curvature -1”.

What exactly does this mean?

First, let us review the definition of a metric, curvature, etc.

A “metric” is a way of measuring distance between two points. For example, the Euclidean metric is how we usually measure distance between two points, as depicted below (Remember the Pythagorean Theorem?).

In this case, we won’t be using the same Euclidean metric, but a hyperbolic metric.

To understand that, we’ll have to understand curvature. Curvature, much like it sounds, is a measure of much an object “curves.”

Let’s look at the three objects we have below.

Figure 1 - Different kinds of curvature.

Figure 2a - A sphere has positive curvature.

Figure 2b - A saddle-shape has negative curvature.

Figure 2c - A plane has 0 or flat curvature.

Figure 3 - The red is the arc going through the circle with endpoints P1, P2, and the yellow is the segment of the diameter going through P1, P2. (We're pretending here that you have two sets of points P1, P2)

Figure 4 - A Euclidean tetrahedron

Figure 5 - A hyperbolic tetrahedron

Figure 6 - Two different hyperbolic knots with different volumes.

Figure 7 - Two different hyperbolic knots with the same volume

The short answer is just that nobody does it by hand because it takes too long.


References:

If we pick a point on the first sphere shape, we see that if we draw cross sections through it, that all of these curve out in the same direction. Therefore, we say that it has positive curvature.

If we pick a point on the second saddle shape, we do the same to get that the curves are in different directions. Therefore, we say that it has negative curvature

Finally, on the last flat surface, if we attempt to do the same, the cross sections all result in lines. Therefore, we say that this has zero or flat curvature.

Now that we’ve established all of this, let’s go back to our original definition:

the hyperbolic knot is a knot with a complement that can be given as a metric of constant curvature -1.

Here, we have established that, for the complement of the knot, we’ll be using a hyperbolic space, particularly the hyperbolic three-space denoted as H3. Essentially, the points in this, instead of being linear coordinates, are points inside a unit ball, and is denoted as:

Now that we have established our space, we need to find our (nonlinear) distance-measuring metric.

Let us choose two points P1,P2 on the unit sphere. Let us draw a circle going through points P1, P2, where parts of the circle going into the unit sphere are perpendicular. We call arc C the arc of this circle with endpoints P1,P2, such that most of it is inside the unit sphere (aka, the red arc in the figure below).

If points P1, P2 lie on a diameter (any segment going through the center of the circle), then, instead, we let C be the part of the diameter between them (refer to the yellow line in the figure above).

We can then say, that there is only one path that fulfills each of these conditions.

Incidentally, in hyperbolic three-space, the shortest path between two points - let’s call it w for the sake of it - turns out to either be a straight line or an arc of a circle (Hint: one of what we just defined).

These turn out to all be geodesics, which are any arc of a circle or diameter  in H3 that is perpendicular to the unit sphere. Incidentally, if points P1, P2 are on the geodesic, then our path w also falls on this. (Think of it as being the “straight line” of hyperbolic space.)

To measure the distance then, we can’t use the same linear measurement, and, instead, integrate along w (shortest path between P1, P2). Make sense, right? More specifically, it’s defined as 

Now that we have our metric, and our space, we can go back and try to understand what exactly this hyperbolic knot (or even hyperbolic knot complement) seems to be. It turns out, that, to create a hyperbolic knot complement, one takes tetrahedron in hyperbolic space and glues them together.

Usually, in our happy Euclidean space, a tetrahedron looks something like this:

With all that we’ve talked about, the general take-away should be that negative curvature spaces have cross sections that curve in different directions. Therefore, when we put our tetrahedron (pyramid) into a hyperbolic space, it turns into something like this:

We end up taking multiple of these tetrahedron, and gluing the faces together inside H3 (the unit sphere), in order to make a knot complement.

So we’ve found out how to make these hyperbolic knots. How exactly do we distinguish between them? Interestingly enough, we can calculate the volume of these knot complements by adding the volume of the tetrahedron we are gluing together.

Turns out, this is actually an invariant of the hyperbolic knots, known as hyperbolic volume, and any two knots with different volumes must be different knots, for example, the knots below.

Sadly, the inverse cannot be said to be true -- there are (a few) knots with the same hyperbolic volume, that are not the same knot.

At this point, you may be wondering, since we’ve calculated the volume of these hyperbolic knot complements, couldn’t we calculate the volume of these knots? Well, the long answer is that you could, by taking the complement and dividing it into n tetrahedra, which you then get n different equations for, and…

While we could go on all day for hyperbolic spaces and hyperbolic knots, that’s it for hyperbolic knots for now! Thanks for listening, and keep up for the next update!

Abyss.uoregon.edu. N.p., 2016. Web. 14 Apr. 2016.

Loki3.com. N.p., 2016. Web. 15 Apr. 2016.

Mathworld.wolfram.com. N.p., 2016. Web. 15 Apr. 2016.

Adams, Colin Conrad. The Knot Book. New York: W.H. Freeman, 1994. Print.