"A knot!" said Alice, always ready to make herself useful, and looking anxiously around her. "Oh, do let me help to undo it!"
- Lewis Carroll, Alice in Wonderland
We all know what a knot is. We use them to tie our shoelaces, to make a lump on the end of a string to stop it going through a hole, or to make a loop on the end of a rope. Sailors use complicated ones for tying up boats, and collections of decorative knots are framed and hung on the wall to add a nautical feel to many a suburban living room.
Mathematicians also study knots, but they have different concerns: which knots can be untied without cutting the rope, how many different knots are there, and how can we tell if two complex knots are the same or different?
A mathematical knot is not that different from a normal knot - it is a curved line which is tangled around itself and then twisted so that the ends meet to form a loop. Like the continuous loops of Celtic knotwork, a mathematical knot has no ends. It is a topological1 entity: you can bend it, shape it, any way you want to, as long you don't cut it. It is the way in which it is tangled around itself that matters. We can tuck a loop into another loop, twist it or pull it out from a hole – all these change the look of the knot but don't change the knot. Any such variations are still considered to be the same knot. Here's an example of a mathematical knot:
This one is known as the overhand or trefoil knot2 and is the simplest possible knot.
Mathematicians also study tangles with more than one line, but they don't call them knots, they call them links. A link consists of two or more knots linked together. We don't say much about links here.
No Free Ends
Why should mathematical knots be confined to ones where the ends are joined together? It's because mathematicians aren't interested in trivial problems. If you have a free end to a knot, then it can be untied by feeding the free end through, so all open-ended knots are topologically equivalent to a straight piece of rope. To make things interesting, there have to be no free ends. There are two ways of doing this: one is to stretch the ends off to infinity and consider the knot to be in the middle of an infinitely long rope. The other is to join the ends together. By some thorny topological manipulations, these two approaches can actually be shown to be the same. So we are justified in looking only at the knots with the ends joined.
The straight piece of rope is as far from a knot as you can get. Joining up the ends to make it into a proper mathematical knot, we get a simple loop, such as a wedding ring or an elastic band. This is considered to be 'unknotted' and is known by the special name of the 'unknot'.
Untying any knot now means manipulating the knot in three dimensions, without cutting, until it has become an unknot. If the knot can be untied in this manner, then it was the unknot all along!
The Classic Problems of Knot Theory
We've already mentioned the sort of questions that mathematicians ask about knots. Now we'll be more specific. There are three main questions:
How can we tell if two knots are the same or different? This is known as the Comparison Problem and is the single most important problem in Knot Theory. Two knots may look different but one may be transformed into another by manipulation, proving that they are in fact the same knot. We would like to be able to examine the knots and perform some calculation or procedure which always gives an answer, determining whether they are the same or different.
The Untying Problem - which knots can be untied to make a simple loop without cutting? The untying problem is connected with the comparison problem – if we can compare a knot to the unknot and show they are the same, then we have proved that the knot can be untied, although we might have no idea how to go about untying it. We would also like a systematic procedure for untying knots that are capable of being untied.
The Classification Problem - is there some way of classifying knots so that we can put them into some order? We'd like a system where we can examine a complicated knot and be able to say that it is knot number 273, for example.
Mathematicians have only partial answers to these problems so far. This entry will try to explain some of the discoveries that have been made.
Knots are three-dimensional objects, but unless you go to the trouble of making models of the knots using string, it is difficult to think of them in three dimensions. It's also difficult to write mathematical articles with bits of string attached, so it is normal instead to use two-dimensional diagrams of the knots.
The normal practice is to draw the knot as a thick black line, with small gaps in the line to show which bits go in front of which at the crossings. Here's the trefoil knot again, drawn in this 2-d style:
The following figure shows the same knot with parts of it stretched or shrunk so that it looks different. If you've a good sense of three dimensions you may be able to see that it is the same knot, or you may need to make a model with string to convince yourself:
The 'figure of eight' knot is a more complicated one:
To avoid all ambiguity, we impose a couple of additional conditions on these 2-d diagrams: we never have three lines crossing each other at the same point; and we never have any section of the knot hiding behind another section - any places where the knot is hidden must be single points only.
A mathematician called Kurt Reidemeister (1893 - 1971) in the 1920s showed that a 2-d representation of a knot can be turned into any other possible 2-d representation of the same knot by performing just three types of operation on the diagram in addition to the bending and stretching already described:
A piece of line can be twisted, introducing a new crossing and a loop, or a twist can be removed.
A loop that lies on top of another line can be moved away from it.
A line that passes near a crossing of two other lines can be moved to the other side of the crossing.
These three moves are known as Reidemeister moves. The following figure shows each of the three possible moves changing a portion of a knot.
If a 2-d knot diagram can be changed into another using only Reidemeister moves, then they are two representations of the same knot.
In order to compare knots, we need some sort of a description of the knot which does not change when the knot is manipulated. It might be a number, a formula or something else. The important thing is that it is not affected by the Reidemeister moves. Such a description which doesn't change is called an invariant. Invariants are highly prized in knot theory. If two knots have different invariants, then they are definitely different knots. On the other hand, if two knots have the same invariant, it doesn't necessarily mean they are the same knot – that would depend on the way the invariant was calculated, but none of the invariants devised so far can allow us to prove two knots are the same. The best we can hope for is that for most knots which are different, the invariants will be different too.
The Alexander Polynomial
The first invariant to be devised was by JW Alexander (1888 - 1971) in 1923. The methods of calculation were difficult, but it was the best that was available.
For each knot, Alexander showed a way that a polynomial, an expression involving powers of x combined together, could be constructed. He proved that if the Reidemeister moves were performed on the knot, it did not change its Alexander polynomial, so the polynomial was independent of whatever way the knot was twisted around. If two knots had different polynomials, then the two knots must be different, since if they were the same, one could be converted into the other using only Reidemeister moves.
The method of calculation of the Alexander Polynomial is too complicated to be discussed in detail here, but the polynomials themselves are simple enough: the Alexander polynomial for the trefoil knot shown above is x − 1 + 1/x. Note that the Alexander polynomial doesn't give you any indication of the shape of the knot. It is just a string of symbols associated with the knot. The Alexander polynomial for the figure of eight knot is 3 − x − 1/x proving that it is a different knot from the trefoil. OK, we knew that, but the same principle can be applied to compare more complicated knots.
The Conway Polynomial
The next development in knot theory didn't arrive until 1970: mathematician John Horton Conway came up with a new invariant which turned out to be a minor variation on the Alexander Polynomial, but was calculated in a much easier way. Conway polynomials can be calculated by doodling on paper, so it is very interesting to amateur mathematicians. We're going to explain the Conway polynomial in great detail. At some point along the way we may lose you, but do try to pay attention, because your understanding of the whole theory will be much improved if you know what the Conway polynomial is all about.
A knot can be cut and joined together to make two new knots, which are generally simpler than the original. This process is continued until we have only knots for which we know the Conway polynomial. These can then be combined using a formula to make up the polynomial of the knot we were looking for in the first place.
First we introduce the concept of orientation: we assign a direction to the knot. At any point on the knot, we put an arrow pointing along the curved line - it doesn't matter what direction we choose to start with3. We can follow this arrow around the line and each point along the line has a direction. You can think of the line as being a hollow pipe full of water. The water is flowing continuously around the knot in one direction.
Now that the knot has an orientation, we can distinguish between two different sorts of crossing on our two-dimensional drawing of the knot. Draw a small circle around the crossing: there will be two lines entering the circle and two lines leaving it. Turn the knot so that the two lines that enter are at the bottom of the diagram and the two lines that leave are at the top. If the line that goes from bottom left to top right goes over the other line, then we call the crossing a positive one. On the other hand, if it goes under the other one, then it is a negative crossing4. Positive and negative crossings are shown here, along with something called a smoothed crossing which we'll explain in a minute:
Flipping and Smoothing
Next, we must understand two operations that Conway calls a flip and a smoothing. Looking at one crossing only, one line is on top and one below. A flip is where we change the lines so that they are the other way around: the one that is on top is put below. To do this, we must cut one line, re-arrange the pieces and join them again. By making this one change at one crossing, we've made an entirely new knot. The flip changes a positive crossing into a negative one and vice versa.
Now the smoothing operation: another way we can re-arrange things at the crossing is to cut both lines and reconnect in a different way so that there is no crossing. Again with a small circle around the crossing, erase everything inside this circle. Now there are two lines going into the circle and two lines coming out. We join up a line going into the circle to the line beside it coming out. Then we join the other inward line to the line beside it going out. We've reconnected the lines so that there is no longer a crossing and we've made a new knot by doing this. The smoothed crossing is the one we've already seen in the figure above.
Defining the Polynomial
Armed with these two operations and some basic algebra, we can define the Conway knot polynomial. The definition probably won't mean much to you at the start, but will become more obvious when you work through the examples. Don't panic! If you take it slowly, it's not actually that difficult, and you can always skip it and study it later.
We use the notation P[X] to mean the polynomial of the knot X.
The polynomial of the unknot is 1: P[unknot] = 1.
For any particular crossing, the polynomial of a knot with a positive crossing minus the polynomial of the same knot with the crossing flipped to a negative crossing is equal to x times the polynomial of the same knot with the crossing removed by smoothing. In mathematical notation:
P[knot with positive crossing] − P[knot with negative crossing] = x.P[knot with smoothed crossing]
Calculating the Polynomial of Two Unlinked Unknots
Consider these knots, labelled A, B, and C. Notice that A has a positive crossing in the middle; B has a negative crossing; C has had the crossing smoothed out, converting it into two unlinked loops. Other than that, the three are identical. We can write the polynomial equation for these three knots as:
P[A] − P[B] = x.P[C]
But also notice that A and B are unknots, so they each have a polynomial equal to 1. If we use these values in the equation, we get:
1 − 1 = x.P[C]
or P[C] = 0. Thus the polynomial of two unlinked loops is zero. This is our first result, and we'll use it in the next section.
Calculating the Polynomial of the Trefoil
The trefoil knot is the simplest 'proper' knot. We'll calculate its polynomial here as a demonstration. Have a look at this figure:
D is the trefoil knot we're interested in. We've numbered two of the junctions: 1 and 2.
E is the same knot with junction 1 flipped. D has the positive version of the junction and E has the negative version. E is an unknot, so it has a polynomial equal to 1.
F is the same knot as D but with junction 1 smoothed. It has been transformed into two linked loops.
G is the same as F, but with junction 2 flipped. F has the positive version of this junction and G has the negative version. G is two unconnected loops, so by the result we calculated above5, its polynomial is 0.
H is the same as F, but with junction 2 smoothed. It is an unknot, so its polynomial is 1.
Now we can do the calculations.
Knots F, G, and H are related by:
P[F] − P[G] = x.P[H]
Putting in P[G] = 0 and P[H] = 1, we get:
P[F] = x
Knots D, E, and F are related by:
P[D] − P[E] = x.P[F]
Putting in P[E] = 1 and P[F] = x, we get:
P[D] = x2 + 1.
This is the polynomial of the trefoil knot. This immediately proves that the trefoil is a true knot and can't be untied without cutting the line - if it could be untied, it would have a polynomial of 1.
Final Comment on the Conway Polynomial
The ultimate goal is an invariant that is always different for two different knots. Conway's polynomial is a long way from this, and was pushed out of the limelight in 1984 when Vaughan Jones produced a much better invariant. Nevertheless, Conway's polynomial was a simple but effective step in the right direction.
The Jones Polynomial
Vaughan Jones was a New Zealand mathematical physicist working in America on an obscure branch of quantum mechanics. He came up with the mathematics first, and then realised it could be applied to knots. He announced his new polynomial invariant in 1984. Unlike Conway's which was just a simpler reworking of the Alexander, Jones's invariant was completely new and much better - it can tell the difference between knots that previously could not be proved to be different using invariants. For example, the Jones polynomials for the trefoil knot shown above and its mirror image are different, proving that they are different knots.
The way by which Jones arrived at his invariant is very complex, but the final product is only slightly more complex than the Conway/Alexander polynomial. It can be calculated by a number of rules which look similar to those used for the Conway Polynomial:
X(unknot) = 1
t-1.X(knot with positive crossing) − t.X(knot with negative crossing) = (t1/2 − t−1/2).X(knot with smoothed crossing)
This looks very daunting, but if you understood about the Conway polynomial, then you'll probably be able to figure that out with a bit of thought. If not, don't worry. The important thing is that it is a fairly simple invariant which is immensely useful in knot theory.
Fired by the simple methods of Conway and the success of Jones, research into polynomial invariants blossomed. Eight mathematicians simultaneously came up with an even better polynomial; probably the only time in the history of mathematics that such an event has occurred! Rather than arguing over who invented it first, six of them decided to publish together, and invented the name HOMFLY for the polynomial, formed from the initials of their six surnames: Hoste, Ocneanu, Millett, Freyd, Lickorish, and Yetter. Two Polish mathematicians, Przytycki and Traczyk, missed the boat due to a slow postal service and their names were not included in the acronym, but later publications sometimes use the name HOMFLYPT to acknowledge their work.
The homflypt polynomial is even better than the Jones polynomial, but it still isn't perfect - there are still knots which are provably different but have the same homflypt polynomial. There is still much work to be done on the comparison problem.
Other Aspects of Knot Theory
What we've seen here is just one part of knot theory, which is a very new topic and is still being actively researched by mathematicians across the globe.
In the 19th Century, Lord Kelvin (William Thomson, 1824 - 1907) speculated that atoms might be knots in spacetime, perhaps knotted lines of force, which it was known could not cross each other. He effectively invented knot theory for this purpose, although for the first 70 years or so, there was no progress other than making tables of different knots. Kelvin's 'Knot Atom' theory came to a dead end with the discovery of neutrons, protons and electrons.
Alexander studied a special type of plaited ropes called 'braids', and showed that every knot can be converted into a braid, that braids can be classified and can be manipulated by a sequence of arithmetical rules. Unfortunately, this is not quite enough to allow a complete classification of knots.
The study of the arithmetic of knots shows that some knots can be broken up into a sequence of smaller knots along the line, while others cannot be broken up any further. These are analogous to prime numbers and the threading operation analogous to multiplication; these basic knots are often known as 'prime knots'. But there is as yet nothing corresponding to addition, so the analogy with numbers is only loose.
There are many other aspects to knot theory and the subject is a thriving field for mathematical research. Who knows what will be discovered or devised in the next few years?