I was in the Army Cadets at school, but my best friend was in the Sea Cadets, and for a year or so I crossed over to join him. This phase of my life came to an end when I failed the oral exam for promotion to the rank of Leading Seaman. The exam was given by a bearded old sea dog — a Captain, no less, with four gold stripes round the cuff of his sleeve to prove the fact. To this day, I remember the question that finished my naval career. Suppose (asked the Captain) I wished to tow a length of timber behind my boat, to season it in the water. What kind of knots would I use to secure it? The answer is at the foot of this column; but I did not know it when asked, got stuck at Able Seaman rank, and have been leery of knots ever since.
Alexei Sossinsky’s little book (it measures only 5 — by 7½ inches) has done much to reconcile me with the world of knots. It is an account of mathematical knot theory, aimed at a non-specialist reader who is willing to take in some unfamiliar, and sometimes quite demanding, algebraic notations. As pop-math books go, I should say that it is at the high end of the range of difficulty for readers who are not mathematicians — closer to Keith Devlin’s The Millennium Problems than to Fermat’s Enigma. Having said that, there are several mitigating factors. The narrowness of the book’s scope helps a lot. Once you have grasped three or four basic ideas, and got into the knotty way of thinking, it is easy to expand your understanding.
I found the author’s style very engaging, too. Alexei Sossinsky is Professor of Mathematics at the University of Moscow, and he originally wrote the book in Russian. The capacity of the Russian language for playfulness and rhetorical subtlety was well advertised to American readers by Vladimir Nabokov. Professor Sossinsky seems to be adept at exploiting these winsome features of his native tongue, and his translator has carried them over very nicely into English. Some of the turns of phrase here are positively Nabokovian. Try this, for example:
Untying a knot often means first making it more complicated (alas, also true in real life). Finally, the functioning of an unknotting algorithm (which is fairly simple but has the disadvantage of futility when it comes to trying to unknot non-unknottable knots) will be explained …
I should love to hear that in the original Russian.
Well, to the substance of the book. You need to understand what mathematicians mean by a knot. The mathematician’s knot is not quite the same as the everyday item. Take a length of string. Tie a knot — or, if you like, several knots — in the string. Now join the two loose ends together, so that the string becomes a closed curve in space, with no ends. That is a knot, in the mathematical sense. The simplest of all knots is the trivial knot, or “unknot,” consisting of a simple circle of string, with no knot in it at all. The next simplest is the trefoil knot, in which you do a single under-and-over before joining the ends of the string. With more complicated knottings, an infinity of different knots can be generated.
Perhaps the most fundamental problem of knot theory arises from the word “different” in that last sentence. Given two very complex knots, how can we determine whether they are, in fact, different? It may be that by jiggling and fiddling with one of the knots, we can make it identical with the other. Then they were “really” the same knot all along. Example: Take an unknot — that is, a simple closed circle of string. By pulling on two diametrically opposite points, “flatten” the circle to an extremely long, extremely thin, oval. Pretending this is just a single open-ended length of string, tie an ordinary knot in it. What you have now does not look anything like a circle. However, by simply sliding the knot open, without cutting the string, you can manipulate it back to the original unknot, to a plain circle. The two configurations are, therefore, from the mathematical point of view, the same.
Given two very complicated knots, how can we determine whether they are the same in this sense? Whether, by merely sliding without any cutting, the one can be transformed into the other? The mathematician’s answer is: we must find an invariant. That is, we must associate with every knot some characteristic mathematical object that is left unchanged by manipulations of the slide-but-don’t-cut type. This object might be a number, or a polynomial expression like x4 – x + 1, or some even wierder denizen of the mathematical zoo. There must be a method to extract this invariant from any given knot. “Here is a knot, Professor Sossinsky. Please tell me its invariant.” “Ah, okay, just a minute. Hmm … hmm … hmm … Right. The invariant is 74.” Then, if you give me two different-looking knots, and they both have the same invariant, I can assure you that they are, in fact, the same knot.
The problem of finding such an invariant is immensely difficult. To this day, in fact, it is not known with mathematical certainty that a “complete” invariant — one that will infallibly identify knots that are the same, and distinguish knots that are different — actually exists. There are a number of nearly-complete invariants, ones that can identify all but some small class of exceptional knots. Sossinsky discusses the Conway, HOMFLY (an acronym for six researchers), and Jones polynomials, and the deep and mysterious Vassiliev family of invariants, this latter probably the best bet for completeness, though we cannot yet prove it. Each type of invariant is introduced through simple concepts and clear diagrams. I think the Jones polynomial, which begins with some considerations in statistical mechanics, will be the pons asinorum of the book for non-mathematicians. It is worth persevering with, though, for after ten pages a very beautiful result is obtained, earning a rare exclamation point from the author, who then says:
God knows I do not like exclamation points. I generally prefer Anglo-Saxon understatement to the exalted declarations of the Slavic soul. Yet I had to restrain myself from putting two exclamation points instead of just one at the end of the previous section. Why? Lovers of mathematics will understand …
We do, Professor Sossinsky, we do.
[Answer to the Captain’s question: A chain of timber hitches.]