(Image: fdecomite, Flickr)
AS MAGIC tricks go, it is nowhere near as spectacular as sawing your assistant in half or pulling a rabbit out of an empty hat. But for mathematicians, there’s nothing in the magician’s repertoire that trumps the amazing vanishing knot. The drums roll, the cymbals crash, and with a triumphant “Hey, presto!” the conjuror pulls an impossibly convoluted tangle of rope into a straight cord.
Anyone who has wrestled with intransigent shoelaces can tell you that the trick does not work for any old knot. The secret of success lies in good preparation: the magician ensures the desired outcome by carefully knotting the rope beforehand, following a well-established procedure. But what is it exactly about the conjuror’s knot that means it can be pulled apart just like that, but the shoelace cannot?
Such questions turn out to have far-reaching significance. Molecules of DNA are often found in Byzantine tangles, and whether or not they can be untied seems to be a crucial factor in determining the likelihood of gene mutation, a driving force of evolution. The mechanical properties of the polymers that are ubiquitous in modern life also depend largely on how their long molecules get knotted. In physics, knots turn up unexpectedly in fundamental areas, from quantum computing to statistical mechanics.
The trouble is, finding a satisfactory answer to the problem of the conjuror’s knot turns out to be decidedly tricky. The branch of mathematics that has developed as a result, known as knot theory, quickly runs up against a conundrum that has tied up some of the best mathematical brains of …
