I brought it out again today to explore a bit more about knots with the boys. The goal of the project today wasn’t depth, but rather just exposure to some basic ideas in knot theory. Knot theory is very simple in some ways an incredibly complex in other ways – I just wanted to show the boys a couple of examples of ideas that help us tell two knots (or links) apart.

We started with a basic idea about links and eventually talked about the linking number. The boys had some nice ideas about how you might tell two links / knots apart:

I felt a bit unsatisfied with the first part of the talk so I decided to show the boys a couple of simple examples of the linking numbers staying unchanged when the links move around a bit:

The next topic we looked at was the un-knotting number of a knot. I used the trefoil knot as the example here. First the boys played around with the knot to convince themselves that it was not the same shape as a circle, or trivial not.

Once they believed it was, indeed, different from the trivial knot, we changed one crossing and found that it was now the same as the trivial knot.

The last part of today’s project was looking at the different types of knots with 0, 1, 2, 3, and 4 crossings. This was just a high level overview and I was hoping to hear some simple ideas from the boys about, say, why all knots with one crossing were the same as the trivial knot.

At the end (and off screen) I asked them to make a version of the knot with 4 crossings. That picture is below:

So, a fun project. I’d love to think a little more about how to make knot theory accessible to kids. It sort of feels as though it is a subject that requires more than just one or two projects, though. Right now I’m not really sure what to do next.

Hey Mike,
I was wondering if you’ve watched any of 3blue1brown’s excellent Essence of Linear Algebra series on YouTube. I feel like it’s a wonderful introduction to the field and parts would be quite accessible to your kids.

When I was learning to tie knots (required for my job at the time!) I had the hardest time with bowline, until it was explained to me that each crossing of the rope was in opposition to the last crossing -easier to see than to explain. If you are still pondering where to go next with this, may I recommend that you get a couple of yards of some lashing rope from a climbing store , then learn some cool knots? Maybe start with tying bowlines, bowlines on a bight, and figure eights on a bight. The bowline is a particularly satisfying because no matter how much tension your apply it holds well but there’s a simple way to break the tension when you want to untie it. The figure 8 on a bight is just a really attractive knot, and, hey, it’s a number 8. Looking forward to seeing what you do next.

## Comments

Hey Mike,

I was wondering if you’ve watched any of 3blue1brown’s excellent Essence of Linear Algebra series on YouTube. I feel like it’s a wonderful introduction to the field and parts would be quite accessible to your kids.

I’ve seen a couple of their videos and really like them. Haven’t watched the linear algebra series yet, though.

When I was learning to tie knots (required for my job at the time!) I had the hardest time with bowline, until it was explained to me that each crossing of the rope was in opposition to the last crossing -easier to see than to explain. If you are still pondering where to go next with this, may I recommend that you get a couple of yards of some lashing rope from a climbing store , then learn some cool knots? Maybe start with tying bowlines, bowlines on a bight, and figure eights on a bight. The bowline is a particularly satisfying because no matter how much tension your apply it holds well but there’s a simple way to break the tension when you want to untie it. The figure 8 on a bight is just a really attractive knot, and, hey, it’s a number 8. Looking forward to seeing what you do next.