Filed under: Mathematics
As I was clearing away some computer cables, I saw a rather unusual knot.
It’s unusual because you don’t normally see such a topologically complicated knot. The most common knot seen in tangled wires is the trefoil knot, the simplest possible knot:
A word on this picture: mathematical knots are always loops, imagine cutting the knot in the picture above and pulling apart the two cut ends to see what it would look like in wires. Click on the picture for more information than you probably want to know about this knot.
So anyway, returning to my cable knot. I wondered to myself: just how unusual is this knot, and which knot is it? This is actually a more difficult question than you might think. In general, the problem of working out if any two knots are the same is very difficult. Mathematically, two knots are the same if you can go from one to the other by pulling and rearranging but without cutting. In fact, determining this in general leads to some of the most abstruse and difficult of modern mathematics, including the subject of my PhD thesis, hyperbolic geometry.
Fortunately, this is a fairly simple knot and I thought it was likely I’d be able to identify it. First thing to do was to make the knot as simple as possible. I used KnotPlot software to input my knot, and it rearranged it to make it a bit prettier:
It also computed the knot’s HOMFLY polynomial for me, which is a good way of identifying knots. For every knot, you can calculate an algebraic expression called the HOMFLY polynomial – this expression is the same however you rearrange it. One way of identifying knots is to compute this polynomial, look up in a table which knots have the same polynomial, and see if you can by hand rearrange your knot to show that it’s the same knot. This worked for my knot.
In the picture above, there are 8 ‘crossings’ (that is, points where one thread crosses another). But, you can rearrange the knot fairly easily so that it only has 7 crossings (the loop at the bottom right that goes through the loop at the top right can be jiggled around a bit so that there are 2 crossings in this area rather than the 3 you can see in that picture. Every knot has a ‘crossing number’ which is the smallest number of crossings you can rearrange the knot to have. The trefoil knot has a crossing number of 3, and it’s the only knot with only 3 crossings. Having rearranged my knot, I knew that it had a crossing number of 7 or less. So I loaded up the tables of all knots whose crossing number is 7 or less using KnotInfo. It turns out there are only 14 of them. Looking up the HOMFLY polynomial I found mine, it’s called ‘7-2‘, which means it’s the second knot in the table with a crossing number of 7. Here is the nicest way of drawing it:
In answer to my original question: there are seven knots in the table which are simpler than this knot, and seven which are equally complex (measured in terms of the crossing number). So in some sense it’s the 8th (equally) likely knot to come across.
If you’re interested in the mathematical theory of knots, try this Wikipedia page.
Oh, and in case you wanted to know, the HOMFLY polynomial for this knot is:
a2 + a2z2 + a4z2 + a6 + a6z2 – a8
Now back to work…