For reasons that escape me, I'm writing a short story that heavily features a branch of mathematics known as knot theory. Wait, there's a branch of mathematics about knots? Why yes, yes there is.

Math is about numbers, duh, but it's also about geometry. Many of the ancient Greek mathematicians did math purely through geometry, in fact, because they didn't have algebra to represent general forms or calculus to deal with infinitesimals. Anywho, there was a point in the 19th century at which studying geometry morphed into studying surfaces, and this led to ideas such as differential geometry and topology. Differential geometry is the math behind Einstein's general theory of relativity, and topology tells us that doughnuts and coffee mugs are the same thing.

Doughnuts and coffee mugs are the same thing? Apparently, yes, because in topology, objects are homeomorphic if you can transform (stretch, squeeze, rotate, twist) one into the other without creating any new shapes or holes. There's a little gif on wikipedia showing that this isn't quite as crazy as it sounds.

Now, there's a sub-branch of topology known as knot theory, which studies circles embedded in space (or spheres embedded in 4-space, etc.). It turns out that some circles are just circles (called the unknot), whereas others are true knots that cannot be transformed back into circles without cutting the knot. Take a rubber band, for example. It's just a circle.

No matter how many times you twist it around and tie it into knots, it's still just a circle that can (theoretically) be undone.

But if you cut the rubber band...

...tie it into a knot, and then reconnect the severed ends (and pretend it's not being held together with tape), then you've created a true knot that cannot be transformed back into the unknot.

You can take that one new knot and twist it all around into different-looking knots, but knot theory says that, as long as you don't cut the rubber band again, it is still fundamentally the same knot, the same way a coffee mug and a doughnut can be fundamentally the same topological space.

So, how'd I write a short story about that? Well, it turns out knot theory has uses outside of rubber bands. In fact, Kelvin kind of got some of the credit for starting this whole knot theory business when he hypothesized that atoms were just knots (vortices) in the aether. But then Michelson and Morley kind of threw a wrench in that whole thing.

There are some modern applications of knots, however. DNA gets itself tied up into knots, and knot theory can help explain how it undoes those knots. Quantum field theory can also be described in a knot-like way, and there might also be quantum computers based on knots.

My story extrapolates this all well into the realm of science fiction, to the point that it would probably piss off the 3 mathematicians who study knots and read short science fiction. But I think it goes in interesting directions. I tie in (ha, not intentional) notions of Buddhist and Celtic endless knots, the Gordian knot legend, and the knot-based number system used by the Inca, known as Quipu. Fun stuff, I hope. I currently have two beta readers attempting to determine whether or not it's ridiculously boring. We'll see.

(Hm. I guess that was a little more than just an update.)

Doughnuts and coffee mugs. Interesting.

ReplyDeleteI'd like to read that story.