## Braided knitting

A few months ago a new sheep, Fernilee, appeared in my flat. (You may remember him from the New Year’s Eve party.) Luckily for him I’m not a very territorial sheep (except for the sofa, which is MINE) but I was a bit put out that he had a lovely hat and scarf and I didn’t. I know that it’s currently summer in Edinburgh, but that isn’t the point.

Seeing as Julia was totally bored after having finished her PhD, I asked if she would make me the scarf to rival all scarves. Not only should it be a warm and functional piece of winter knitwear, but it should embody some sort of mathematical principle so that I can continue inspiring my followers wherever I go. We brainstormed a few ideas: having a stripey scarf where the number of rows of each colour followed the digits of pi (e.g. see here), or having a hidden mathematical image knitted into the scarf (so-called illusion knitting), or having a braid pattern using cabling. The first idea seemed too easy and the second one quite hard, so we decided on a braid.

The braid pattern we chose

Julia had never tried cabling before and wasn’t sure how to design a pattern from scratch, so we decided to find a ready-made pattern for our first braiding attempt. Eventually we decided to go with this one modulo some modifications – changing the border to seed stitch to make it easier, and adjusting the cabling pattern to make the braid alternate (over-under-over-under). It’s a 6-stranded braid with no special mathematical properties (that I can see). In particular, it is definitely not the same as this commutative braid which my officemate Patrick and my old supervisor Andrew are working with.

Being mathematicians (and knot theorists!) definitely helped us to figure out how the cabling pattern worked. In a mathematical braid there are a number of strands running parallel to each other, and every now and then two adjacent strands are allowed to cross. If the strands are labelled 1 to n, then the crossings are denoted by a sequence of numbers, where i means that strand i crosses OVER strand i+1, and -i means strand i crosses UNDER strand i+1. For example, the braid below would be denoted by (1,-2,1,-2):

The braid 1,-2,1,-2

This is in some ways very similar to knitting braids, because in a knitted braid only adjacent strands are allowed to cross. The cable pattern denotes whether the crossing is OVER or UNDER by using F (‘front’) and B (‘back’). The first obvious difference between the maths braids and the knitted braids is that knitted strand-crossings are allowed to occur simultaneously between non-interfering strands. E.g. strands 1 and 2 can cross at the same time as strands 3 and 4 do. Mathematically it makes no difference, but aesthetically it is more pleasing to have simultaneous strand crossings.

The next similarity between maths and knitting is how we ‘add’ braids together. It is simply by putting them side-by-side, the second braid following on from the first. Similarly, the knitting pattern only gives the first 16 rows – the first ‘block’ – and then the braid is continued by placing these blocks next to each other along the scarf.

Braid theory can design paths for these robots so they don't crash into each other

Mathematically, braids are interesting because their addition has a lot in common with numbers. There is a ‘zero’ braid which does nothing when added to another braid – it is the braid with n strands running in parallel. There are also ‘inverse’ braids, which are like negative numbers in the sense that when you add a braid to its inverse you get the zero braid back. (Can you figure out the inverse braid to the (1,-2,1,-2) braid drawn above?) This additive structure makes the collection of braids into a group, and the braid group is of great interest to a lot of people in the world right now! Engineers use them for motion planning in robotics, cryptographers use them to design new codes and computer scientists are using them to design quantum computers.

Braids can also be turned into knots by joining the strands at the end of the braid back to the beginning. I think if I make another braided scarf I shall try to encode the braid for the knot 12a631, which is the only knot in my thesis where I couldn’t decide if it was slice or not.

I will end this post by showing you some pictures of how Fernilee reacted to my beautiful new scarf. You can see that he was quite jealous!

Fernilee looks on from behind as I try on my new scarf...

We have a little chat...

He inspects the scarf and admires the mathematical braid

We make friends and play a game of Ninja Sheep!

### 4 responses to this post.

1. Great post, I’d never given much though to how braids can be applied ‘in real life’ but now I have an idea and can see how useful they are. *toddles off to see if she can work out the braid notation for her current knitting project* Could a knitting pattern be written in that notation? What would you’re celtic scarf come out as?

2. There’s an entire chapter on knitting, cabling, and braids in the book Making Mathematics with Needlework (2007).