## Why not knot wire?

I have been thinking quite a bit recently about ideas of knotting and weaving. There will probably be another post on the theme soon. As a mathematician it brought me straight back to Knot theory, I love looking at the strange images that appear on the blackboards in the lectures and offices of topolgists, many of which contain knots. This video lecture from Elvis Zap is a classic example (even if you cannot follow, just sit back and enjoy the drawing!). Not to forget the beautiful uses knotted designs have been put to outside of mathematics.

At some point during this I needed something made out of metal, and decided to bend some wire into a trefoil. It was satisfying, so I though I would look online to see if I could find collections of physically made knots. These were surprsingly hard to find. There were plenty of examples to be found (even the Museum of Mathematics‘s famous knotted bagel), but I could not find any systematic collections. So I decided to make my own, using this knot zoo for reference. Here are the knots that can be drawn with seven crossings or less, using Conway’s tangle notation:

It was great fun making the knots and I encourage anyone who studies them, even idly, to have a go. I felt the knots themselves come alive in my head as I made them. I started to think how the knots could be put together out of sections of twists, further study of this lead me to tangles and Conway’s notation. You might notice that this came late as the written labels on the knots are the more commonly used Alexander–Briggs notation. That is a lot less satisfying as after the number of crossings the numbers to not refer to the properties of the knots.

In addition making sure that the wire holds naturally in shape without touching itself is great for 3d intuition. One thing that struck me as I started to bend the wire was how 2 dimensional most knot images are. The crossing number is a classic example of this as it is a 2d not a 3d property. There are, of course, good reasons for this both in design and exposition, but it was interesting feeling how the knots changed as you allow to move more freely. Of course this had some issues when I came to present the knots here, of course in 2d (I hope I managed to get all the pictures so spare crossings are easy to remove!). A video might work slightly better:

I love those knot sculptures! Now that you mention it I realise that knots are very 2d in my head. Probably because I would not be looking at them without the material touching each other,i.e. a string (perhaps my strongest image of a knot is the horrible knot on my shoe laces that is impossible to untangle). Thanks for bringing knots into a more interesting context!

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