A Personal Story about Knots

Gavin Wraith

What turned me on to mathematics

At school I knew a boy called Pasternak-Slater. I believe he was the nephew of Boris Pasternak. One day he was doodling on a piece of blotting paper, writing down the numbers from 1 to 5 in a different sequence. Let us say he wrote down 25134. Then he would write down the positions of the digits, first the position of 1 (which is in the third place), then that of 2 (which is in the first place) and so on, getting 31452. He noticed that if he repeated the process on the newly obtained sequence he got 25134, the sequence he started with! This intrigued me, and I tried to explain why. First of all, it clearly had nothing to do with the number 5, because exactly the same phenomenon can be observed however many numbers one takes. I then argued that the process of obtaining the new sequence was using the firstness of the digit 1, the secondness of the digit 2, and so on. If I had used some other symbols, say %$@&!, I would be stuck, because without assigning some natural ordering to them there would be no way of getting a new sequence. So the natural sequence 12345 must be playing a special role, I argued. Then I noticed that if I wrote down a sequence, then the natural sequence, and finally the new sequence obtained from the first by the Pasternak-Slater process, something interesting was staring me in the face:




I saw that whenever in the first two rows one digit appeared beneath another, then the same thing was true for the last two rows.




This suggested a way of composing sequences of any set of symbols without the need for some natural ordering of them.


I started to use names for sequences. The composition of two sequences was to be got by the same rule that I had observed, so that if a symbol Fred in stood in the same position as Harry in then Fred must stand in in the same position that Harry does in . I called my newly discovered algebra sequentials. I filled notebooks with computations. If one writes N for 12345 (N for Natural) then the Pasternak-Slater process becomes composition with N on the right hand side and the phenomenon he observed becomes the rule that . I soon saw that


was a general rule of sequentials.

Two other general rules I found were



and these three rules I dubbed the basic rules of sequentials. Then I realized that the same rules applied to points in the plane, if by PQ one meant the point got by travelling from P to Q and then continuing an equal distance on the other side. The rules (PQ)Q = P and PP = P are now trivial, but (PQ)R = (PR)(QR) is a bit more interesting.

As a consequence of my curiosity about these things I decided to take the option of spending more hours studying mathematics, at the expense of fewer hours studying French. When I left school and arrived at Cambridge I did the Mathematics Tripos. A course on group theory and permutations made me realize a bit more clearly what my sequentials really were. If one associates to a sequence x the permutation that coverts N to x then sequential composition is explained by the rule


In the same way, if one associates to a point P a vector v(P) from some base point O to P, then one has v(PQ) = v(Q) - v(P) + v(Q). It became clear, in fact, that if one took any group G and defined an operation # on the elements of G by the rule


then the sequential laws all held. I also noticed that that two of them (S2, S3) held if one made the alternative definition


I soon got to know John Conway, who renamed sequentials wracks (a wry comment on my name) and we spent many hours working together or exchanging letters to study them in terms of groups.

Chasing wracks gave me a taste for algebra.


Many years later, at the University of Sussex, my colleague Roger Fenn, whose office was next to mine, strode in to my office and wrote

(X Y) Z = (X Z) (Y Z)

on my blackboard. Ever seen anything like that?, he asked. I replied that indeed I had, and I told him the story of the previous paragraph, and put into his hands some rather dog-eared letters from John Conway.

Roger explained that the system he needed was slightly more general than the wracks that John Conway and I had considered. He wanted systems that satisfied the right-self-distributive law (xy)z = (xz)(yz) but not necessarily the other two wrack laws; instead they should satisfy the condition that for each y and z there should be a unique x such that xy = z. He called these more general algebras racks to distinguish them from wracks. Actually, wracks had been reinvented quite independently under the name of involutive quandles during the intervening years by D.Joyce, in J. Pure and Applied Algebra 23 (1982), 37 - 65. A quandle is a rack that satisfies the law xx = x.

Roger was off to a university in Spain for six months. When he returned the first thing he said was: You know, those racks we talked about, they provide a complete classification for framed links in 3-manifolds.

A knot is a smoothly embedded circle. A link is a smoothly embedded union of circles; that is, lots of knots tangled with each other.

Here is a picture of a link with 5 components, blue, red, yellow, orange and purple. In fact, this link is trivial; it can be disentangled into 5 separate unknotted loops. If instead of string you use ribbon, you get the notion of a framed link. To be precise, a framed link is an embedding of a union of circles with a nonzero section of the normal bundle of the embedding.

One is interested in knowing when one link can be deformed continuously into another. A link invariant is a method of calculating a mathematical gadget of some kind from a link, so that if two links are deformable into each other the gadgets are the same (or at least isomorphic). The idea is to reduce the effort of checking that two links cannot be the same to that of checking that the gadgets are not the same, which may be an easier task.

One of the earliest link invariants to be studied is the fundamental group of the complement of the link. This can be defined as follows:

Choose a point P not on the link. Consider paths that start and end at P but which, though they may wind through it, never touch the link. Two such loops are considered the same if they can be deformed one into the other without moving or touching the link. Such loops can be composed, and the inverse of a loop is obtained by travelling along it backwards. This gives a group structure, with unit element the loop that simply stays at P.

There is a well-known procedure for finding this group. Lay the link flat on a sheet of paper to get a link diagram, so that where two segments of string cross you can distinguish one as the overcrossing and the other as the undercrossing:

Choose an orientation for the link; that is, fix a direction of travel round each component. Label all the string segments that are overcrossings, from where they emerge to where they disappear under other segments. Identify an outgoing overcrossing segment (blue) label with the conjugate of the ingoing segment (green) label by the label of the overcrossing segment (red) that passes from right to left (or its inverse if from left to right).

Take the free group generated by the labels subject to these identities, one such for each crossing. The element given by a label X can be identified with a loop starting at P that winds once round the segment that X labels, in the appropriate sense, and then returns by the same route it came back to P.

This group is a link invariant. Unfortunately it is not a complete invariant; that is, there are different links that give rise to the same group.

Joyce's idea was to construct not a group but a quandle, using the picture

Why does this give a link invariant? To verify that a construction gives a link invariant it is enough to check that it does not change under the Reidermeister moves:

This picture shows clearly the geometry behind the self-right-distributive law.

What do the elements of this quandle look like? They can be interpreted as equivalence classes of lassoos; that is, paths from P that end in a little curtain-ring that can slide on the link. There is a function from the quandle to the group taking composition to conjugation, got by allowing the curtain-ring to swell up into a loop. The importance of the quandle of a link is that it provides a complete link invariant.

You can appreciate the difference between lassoos and loops when you realize that a loop can slide over knotted lumps whereas curtain-rings cannot.

A good, Cheshire cat, way of thinking about racks and quandles is that they are what you get by taking groups, throwing away the group composition but retaining conjugation.

Suppose you have a group G with a right action on a set A. A particular example is got by taking A to be the set of elements of G and the right action to be conjugation. Let us call this example @G. Its G-orbits are the conjugacy classes of G. Whenever we have a right G-set A, together with a G - equivariant function | |: A -> @G we can make A into a rack by defining x#y to be x |y|. In fact this construction is what we have in the case of links; G is the fundamental group at P of the complement of the link, and A is the set of lassoos at P that can slide on the link. The right action is composition of a loop with a lassoo, and | | is the function that explodes the curtain-ring to a loop. Various names have been given for these objects: crossed G-sets, automorphic sets, crystals, . . . . Freyd, Crane and Yetter pointed out that the category of right G-sets over @G has a natural braided monoidal structure.

If we have | | : A -> @G and | | : B -> @G then we may define | | : AxB -> @G by the formula |(a, b)| = |a| |b|, defining a monoidal structure on G-Set/@G, with unit object a singleton set {e} with |e| the unit of G.

Then the functions t : AxB -> BxA given by t(a, b) = (b, a |b|) define a braiding of this monoidal category.

Given a link, whose complement has fundamental group G, each component of it gives an object in this category - the elements of the G-set are the lassoos sliding on the given component. They will generate a monoidal subcategory of G-Set/@G with particular geometrical significance.