Topology 3: Knots Galore!
I’ve recently gotten my drawing tablet and it’s made me way more motivated to work on this blog, so I guess we’re cranking out a Part 3 at 8:21 am!
Today we’re going to talk about knots. These are a fun topic for many reasons. They’re low-dimensional, so we can visualize them easily; they’re very intuitive, since most of us have played with strings at some point; and they have a surprisingly rich theory that starts at a fairly basic level!
But we had an issue last time. All knots are homeomorphic. Which is no good. Homeomorphism is what mathematicians call an equivalence relation, it’s a way of saying “two things are equal”. We can choose other equivalence relations. You’ve met equivalence relations throughout your life, where things aren’t equal but are the same. Similarity or congruence of triangles, matrices with the same diagonal matrix, even reduction of fractions (4/3 = 8/6). So far we’ve used homeomorphism as our equivalence relation. Recall that the 3 permitted steps were
- Stretching
- Compressing
- Cutting but only if you reglue correctly
This last point kind of messed us up for knots. While you can’t stretch or compress a lot of knots into each other, cutting means all knots are trivial (unknotted)! So let’s maybe define a new equivalence relation with only the first 2. Without going into the details, this relation is called ambient isotopy, and the two knots are said to be ambient isotopic. This is the relation or “sameness” that agrees with your intuition! A couple of examples below.
This is the “unknot”, the simplest knot which basically isn’t knotted. You can see that the knot on the right is just a twist of it, and you can untwist to get back to the unknot.
Here is the simplest knotted knot, the trefoil! Here’s an interesting fact, I usually draw the trefoil as on the left. But if you change every crossing (i.e. switch which strand is over and under), you still get a trefoil, but these are not ambient isotopic! These are the left and right handed trefoil (I’m not sure which is which) and if you try playing with string, you’ll see that you can’t turn one into another. Knots can display chirality (or handedness!)
Regardless, we will treat the trefoil as one kind of knot (unless handedness matters), and clearly it is not trivial (i.e. not ambient isotopic to the unknot, or in layman’s terms, it cannot be unknotted)
In the last example we tried changing all crossings. What if I just change one? If you look at the crossing change here on the trefoil, you see that you get something trivial! The knot on the right is just the unknot, but looped on itself twice, like when you make two loops out of a rubber band.
Intuition works fine for small knots, but how can you tell if something more complicated is knotted?
We need to develop some tools. Let’s first try understanding ambient isotopy better. We’ve given an intuitive definition but how do we actually knot if there is an ambient isotopy between knots? Reidemeister (and others) discovered that two knots are ambient isotopic if and only if they can be related using a sequence of the following three Reidemeister moves.
These moves assert that moving performing this move either forward or backward leaves the knot unchanged. You can think of these of the basic moves to generate every ambient isotopy. The moves are very intuitively obvious.
Type I — “The Loop/Unloop”
You can introduce a loop anywhere along a knot just by twisting that area, or unlooping. This doesn’t change the knot! That’s sort of what I did with the unknot above if you think about it, maybe drawing it this way will help see that.
Type 2 — Overlapping strands
Again, a really intuitive knot move, this is barely a move! If you have two strands nearby, you can just lay them on on another without changing the knot. Here’s an example, again with the unknot.
Type 3 — Pull Through
This is the most complicated looking but again is very obvious. You can always pull an unrelated strand (green) under a crossing (orange). It lies under the plane of the crossing so it doesn’t matter if it’s above or below. Example below!
Yet again, weird versions of the unknot. In fact the right is what was the “trefoil with one crossing changed” from above!
Hopefully all three of these moves seem intuitive. These are all things you can do with a closed piece of string or with a rubber band.
So now we know how to understand knots and when they are the same. Recall, algebraic topology now tries to build invariants! Let’s try a few.
Attempt: Crossing Number
This seems like a good place to start? One of the easiest things to count is the number of crossings. But unfortunately, this is not an invariant! Just look at the first unknot I drew. The left has no crossing and the right has 1! In fact, the Type I move allows us to introduce as many crossings on a random strand as we want (just make twists everywhere along the knot). Let’s try something a little lamer then.
Refined Attempt: Minimum Crossing Number
This is a bit of a cheat. If you take a knot, you will have a “version” (we call the different versions diagrams) that has the lowest number of crossings. So you can classify knots by what their lowest crossing number is. The reason this is a cheat and not helpful is because you need to ambient isotope the knot to the correct version first, which is a very hard problem, how do you know which moves to perform? “Trivially”, two knots with different minimum crossing numbers are not ambient isotopic because if they were, they’d have the same minimum crossing number since you could ambient isotope both to the simples diagram .On the plus side, this gives us a dictionary of knots! We can organize knots by minimum crossing number, which is how you get the table below.
These are the so called “prime knots”. You can get a lot of other knots by “connect summing” (exercise from Topology I) these knots! The knots are named by the number of crossings and then the subscript is just an index that refers to that knot. Again, we treat mirror images as the same knot.
Even Better: Unknotting Number
Remember when we switched a crossing on the trefoil and made it an unknot? That’s how we unknotted it. Well, if you have a knot in the version with minimum crossings, you can count how many crossings you have to switch before you reach the unknot! We basically saw for the trefoil it was 1. This also suffers from the issue above that you have to get to this “version with minimum crossings” first.
Fun Invariant: Tricolourability
A (tri)colouring of a knot is painting each strand of a knot with one of three colours, subject to the following rules
- At least two colours must be used in the whole knot.
- At a crossing, either all three strands must be the same colour, or all three must be different.
Here is an example of a tricolouring below. Note that at any crossing there are three parts, one over strand, and two parts of the understrand. These are the “three strands” referred to in point 2 above.
The amazing fact is that tricolourability is an invariant! In other words, if two knots are the same, they are both either tricolourable, or they are both not. And conversely, if one knot is tricolourable and the other is not, they are not ambient isotopic. Note for example that the unknot is not tricolourable since you have to use at least two colours!
It’s actually easy to show that tricolourability is an invariant. All you have to show is that if you can tricolour before a Reidemeister move, you can tricolour after! Below is an illustration
Exercise: The Figure 8 knot (below) is not tricolourable.
Hint: Just try colouring the strands! [There are 4 strands, you’re trying to use 3 colours, so you’ll have to use the same colour at some crossing, but if you use the same colouring at a crossing, then they all have to be the same colour! Which makes 3 strands the same colour, and one different, but then at another crossing you’ll have 2 of one kind and one of another which is not allowed!]
So, we have shown:
Big Result of this Section: The Figure 8, Unknot, and Trefoil are all different knots.
- The Figure 8 and Trefoil are different since the Figure 8 is not tricolourable but the Trefoil is
- The Figure 8 and Unknot are different because the Figure 8 has an unknotting number of 1.
- The Trefoil and Unknot are different since the Trefoil is chiral (has handedness) but the unknot doesn’t.
Generating Knots
So how do you go about finding knots in the wild? In Topology I, we showed that you can get surfaces by repeatedly connect summing tori to get many-holed tori. Well, we can do something similar.
Connect Sum (Again)
To get the Connect Sum (also called knot sum) of two knots M and N (denoted M#N), just cut a small piece off of both and join them along the cut strands.
You can build a lot of knots by summing all the prime knots!
Torus Knots
Here’s a fun way to make knots. Take a string and wrap it around the torus, p times longitudinally and q times meridianally (I don’t actually know which is which so just think p times around the centre hole and q times around the tube).
This one’s a little hard to draw but it really is wrapping string around a torus and closing a loop.
For example in the one I drew, p = 1, q = 6. So this is the (1,6)-Torus Knot.
Amazing Fact: Torus Knots are really easy to classify and tell if they’re ambient isotopic!
1. The (p,q)-Torus Knot is ambient isotopic to the (q,p)-Torus Knot
2. A Torus Knot is only trivial (i.e. is the unknot) if one of p or q is 1 or -1. (This just looks like a loop on the torus either meridianally or longitudinally). This means that every other torus knot is non-trivial!
3. All Torus Knots are chiral (have handedness) and in fact, the (p,-q) knot is the mirror image of (p,q) knot.
4. Each pair of co-prime integers determines a different knot. Basically, this means always reduce p/q to the lowest fraction, and that determines a unique knot! So (2,3) is not the same as (2,5) or (2,7)!
Besides the unknot, you’ve actually met another torus knot before! The (±1,±3) and (±3,±1) knots are actually the trefoil!
The study of knots is much more richer than this, but though this would be a fun introduction. Until next time!