A Summer 2015 Mathematics A To Z: knot
It’s a common joke that mathematicians shun things that have anything to do with the real world. You can see where the impression comes from, though. Even common mathematical constructs, such as “functions”, are otherworldly abstractions once a mathematician is done defining them precisely. It can look like mathematicians find real stuff to be too dull to study.
Knot theory goes against the stereotype. A mathematician’s knot is just about what you would imagine: threads of something that get folded and twisted back around themselves. Every now and then a knot theorist will get a bit of human-interest news going for the department by announcing a new way to tie a tie, or to tie a shoelace, or maybe something about why the Christmas tree lights get so tangled up. These are really parts of the field, and applications that almost leap off the page as one studies. It’s a bit silly, admittedly. The only way anybody needs to tie a tie is go see my father and have him do it for you, and then just loosen and tighten the knot for the two or three times you’ll need it. And there’s at most two ways of tying a shoelace anybody needs. Christmas tree lights are a bigger problem but nobody can really help with getting them untangled. But studying the field encourages a lot of sketches of knots, and they almost cry out to be done out of some real material.
One amazing thing about knots is that they can be described as mathematical expressions. There are multiple ways to encode a description for how a knot looks as a polynomial. An expression like contains enough information to draw one knot as opposed to all the others that might exist. (In this case it’s a very simple knot, one known as the right-hand trefoil knot. A trefoil knot is a knot with a trefoil-like pattern.) Indeed, it’s possible to describe knots with polynomials that let you distinguish between a knot and its mirror-image reflection.
Biology, life, is knots. The DNA molecules that carry and transmit genes tangle up on themselves, creating knots. The molecules that DNA encodes, proteins and enzymes and all the other basic tools of cells, can be represented as knots. Since at this level the field is about how molecules interact you probably would expect that much of chemistry can be seen as the ways knots interact. Statistical mechanics, the study of unspeakably large number of particles, do as well. A field you can be introduced to by studying your sneaker runs through the most useful arteries of science.
That said, mathematicians do make their knots of unreal stuff. The mathematical knot is, normally, a one-dimensional thread rather than a cylinder of stuff like a string or rope or shoelace. No matter; just imagine you’ve got a very thin string. And we assume that it’s frictionless; the knot doesn’t get stuck on itself. As a result a mathematician just learning knot theory would snootily point out that however tightly wound up your extension cord is, it’s not actually knotted. You could in principle push one of the ends of the cord all the way through the knot and so loosen it into an untangled string, if you could push the cord from one end and if the cord didn’t get stuck on itself. So, yes, real-world knots are mathematically not knots. After all, something that just falls apart with a little push hardly seems worth the name “knot”.
My point is that mathematically a knot has to be a closed loop. And it’s got to wrap around itself in some sufficiently complicated way. A simple circle of string is not a knot. If “not a knot” sounds a bit childish you might use instead the Lewis Carrollian term “unknot”.
We can fix that, though, using a surprisingly common mathematical trick. Take the shoelace or rope or extension cord you want to study. And extend it: draw lines from either end of the cord out to the edge of your paper. (This is a great field for doodlers.) And then pretend that the lines go out and loop around, touching each other somewhere off the sheet of paper, as simply as possible. What had been an unknot is now not an unknot. Study wisely.