Professor Stewart's Hoard of Mathematical Treasures (24 page)

However, the innocuous-looking series
is a different matter. Bracketed like this:
it reduces to 0 + 0 + 0 + . . ., which surely must be 0. But bracketed like this:
it becomes 1 + 0 + 0 + 0 + . . ., which surely must be 1. (The extra + signs in front of the brackets are there because the minus sign does double duty: both as an instruction to subtract, and to denote a negative number.) No less a figure than the great Euler tried the same sort of trick that we used to sum the first series, letting s be the sum and manipulating the series to get an equation for s. He observed that
and argued that s =
.

This is a nice compromise between the conflicting values 0 and 1 that we’ve just found, but at the time Euler’s suggestion
just muddied the waters further. And they were already fairly murky. The first satisfactory answer was to distinguish between convergent series, which settle down closer and closer to some specific number, and divergent ones, which don’t. For instance, successive steps in the first series give the numbers
which get ever closer to 2 (and only to 2). So this series converges, and its sum is defined to be 2. However, the second series leads to successive sums
which hop to and fro, but never settle down near any specific number. So that series is divergent. Divergent series were declared taboo, because they couldn’t safely be manipulated using the standard rules of algebra. Convergent series were better behaved, but even those sometimes had to be handled with care.

Much, much later it turned out that there are clever ‘summation methods’ that can assign a meaningful sum to certain divergent series, in such a way that appropriate versions of the standard rules of algebra still work. The key to these methods lies in the interpretation placed on the series, and I don’t want to dig into the rather technical ideas involved, except to record that Euler’s controversial
can be justified in such a setting. In astronomy, another approach led to a theory of ‘asymptotic series’ that can be used to calculate positions of planets and so on, even though the series diverge. These ideas proved useful in several other areas of science.

The first message here is that, whenever a traditional concept in mathematics is extended into a new realm, it is worth asking whether the expected features persist, and often the answer is ‘some do, some don’t’. The second message is: don’t give up on a good idea, just because it doesn’t work.

The Great Whodunni, with the assistance of Grumpelina, produces a length of soft rope from thin air and ties a knot in it. A little further along, he ties a second knot. Holding the two free ends in each hand, he gives the rope a shake - and the knots disappear.

Mathematically, it’s obvious, of course. The second knot must be the anti-knot of the first one. You just tie it so that all the twists and turns cancel out. Right?

Wrong. Topologists know that there is no such thing as an anti-knot.

To be sure, there are very complicated knots that turn out not to be knotted at all. But that’s a different issue. What you can’t do is tie two genuine (un-unknottable) knots in the same piece of rope, clearly separated from each other, and then deform the whole thing into an unknotted piece of rope. Not if the ends of the rope are glued together or otherwise pinned down so that the knots can’t escape.

Not only do topologists know that: they can prove it. The first proofs were really complicated, but eventually someone found a very simple proof. Which is completely outrageous. You probably won’t believe it when I show it to you. Especially not when we’ve just been exposed to the paradoxical properties of infinite series.

A mathematician’s knot is a closed curve in space, and it is genuinely knotted if it can’t be continuously deformed into a circle - the archetypal unknotted closed curve. Real knots are tied in lengths of string that have ends, and the only reason we can tie them at all is because the ends can poke through loops to create the knot. However, the topology of such ‘knots’ isn’t very interesting, because they can all be unknotted. So mathematicians have to redefine knots to stop them falling off the ends of the string. Gluing the ends into a circle is one method, but there’s another one: put the knot inside a box and glue the ends to the walls of the box. If the string stays inside the
box, the knot can’t escape over the ends. (The box can be any size and shape provided it is topologically equivalent to a rectangle; in fact, any polygon whose edges don’t cross is acceptable.) The two approaches are equivalent, but the second is more convenient for present purposes.

Two knots tied in boxes . . .

. . . and how to add them.

If you tie two knots K and L in turn along two separate strings, then they can be ‘added’ by joining the ends together. Call the result
K
+
L
. The unknot, a straight string without a knot, can sensibly be denoted by 0, because
K
+ 0 is topologically equivalent to
K
, which we can write as
K
+ 0 = K by employing the equals sign to indicate topological equivalence. The usual algebraic rules
can also be proved; the second one is easy, the first requires more thought.

Now we can see why Whodunni’s trick must, indeed, be a trick. In effect, he appeared to tie two knots
K
and
K
* that cancelled each other out. Now, if two knots
K
and
K
* cancel, then

I’m tempted to replace
K
* by -
K
, because it plays the same role, but the notation gets a bit messy if I do.

The outrageous idea is to consider the infinite knot
Bracketed like this:
we get 0 + 0 + 0 +. . ., which in topology as well as arithmetic is equal to 0. But bracketed like this:
we get K + 0 + 0 + 0 + . . ., which in topology as well as arithmetic is equal to K. Therefore, 0 = K, so
K
was not a genuine knot to begin with.

In the previous item, we saw that this argument is not legitimate for numbers, and that’s what makes the proof seem outrageous. However, with some technical effort it turns out that it is legitimate for knots. You just have to define the infinite ‘sum’ of knots using ever-smaller boxes. If you do that, the sum converges to a well-defined knot. The manipulations with brackets are correct. I don’t claim that’s obvious, but if you’re a topologist it pretty much is.

Tying a wild knot inside a triangle formed from an infinite sequence of shrinking trapezoidal boxes.

Infinite knots like this are called wild knots, and as the name suggests they should be handled with care. A mathematician called Raymond Wilder invented an especially unruly class of knots. You can guess what those are called.

Smith and Brunnhilde had penetrated to the inner sanctum of the Solar Temple of Psyttakosis IV, overcoming various minor obstacles on the way, such as the Pit of Everlasting Flame, the Creepy Crocodile Corridor, and the Valley of Vicious Venomous Vipers. Now, panting slightly from their exertions, they stood at the edge of the Temple Plaza - a square array of 64 slabs, four of which were decorated with a golden sun-disc. Behind them, the only entrance had been closed by a shining disc of solid gold with the weight of a dozen elephants.