A sallow, well-groomed businessman has adopted a defensive posture and avoids contact with others. A shabby drunk tries to strike up conversations without success. A survey would show that most of us are concerned only with getting onto the bus as soon as possible, dropping into a seat and getting some sleep.
With the demise of Trailways in the mid 80's, the bus is probably a Greyhound, though it could also be Peter Pan, or Peerless, or Connecticut Limousine, or Bonanza . No matter: the trajectories presently to be subsumed apply to all buses of a certain height, length, width and construction.
Our bus came in half an hour behind schedule; then it had to be withdrawn from circulation for cleaning and minor repairs. A new driver appeared; there were further delays because of paper work, perhaps some misunderstandings with the central headquarters in Dallas. Let's assume that it's an hour and a half after the scheduled departure time. Finally a dulcet voice, tinged with doomsday, comes over the loudspeaker and announces that the bus is ready for boarding.
As a collective sigh of relief, so strong that it is virtually visible, rises from the crowd, our bus, newly serviced, straining in harness like a well-groomed horse eager for the journey, emerges from the shed and enters its stall. The driver opens the door from inside, steps down onto the loading dock and begins collecting tickets.
This is the initial terminus of this particular route, there are no persons in transit waiting to reclaim their seats. The bus is void, ripe for invasion. The colloidal mass of passengers whips into a linear spine and begins to slouch forward into the bus.
Under this set of initial conditions, Stage A comes to an end when the first wave of 23 passengers have occupied the window seats in all the ranges.
This recognition always occurs in a tiny region which we identify as the "critical point", or "C.P." for short , and is usually located between 0.75 and 0.8125 of the way down the full length of the bus . Beyond it lie two ranges on the left, two on the right, and the 3-seater at the back still remaining, for a total of 11 places, 5 of which, from stage A, are filled.
The first individuals to reach the critical point may now want to turn back, only to discover that it is too late to do so: a hoard of angry passengers blocks their way. This compels those now at the head of the line to take some kind of immediate action.
They will therefore begin settling into whatever seats they find immediately at hand, starting with the ones just beyond the critical point, up to the back wall. Observe that the seating will tend to unfold in reverse order: the seats at the critical point being occupied first, then the next, and finally the two worst seats in the bus, in the 3-seater. Stage B now comes to an end when the upper quarter of the bus is totally occupied. Between it and the contribution of stage A, 29 seats have now been filled.
A Link is a collection of knots. In a one-component link, the knots are inseparable. The Ballantine Rings are an excellent example of a 1-component link: each pair of loops is "unlinked", but the 3 of them together cannot be separated:
The simplest non-trivial knot is called the Trefoil , and looks like this:
A Braid is a collection of N strands in 3 dimensional space extending from a set of N initial points to N terminal points. As a braid can also be seen as a knot that has been cut through at a certain place, and since a knot can always be formed out of a braid by connecting up the initial points with the end points in a certain way, Braid Theory and Knot Theory are interchangeable, although there is considerable difference in the approach.
Although real knots exist only in 3-dimensional space, it can be mathematically demonstrated that all knots are equivalent to their shadows, or 'knot diagrams' projected onto a plane, provided one labels the under- and over-crossings at the self-intersections of the diagram , and takes care that the projection of one intersection does not fall directly on top of another. A single object casts many shadows: the same knot can project into many diagrams. The equivalence of all the diagrams of the same knot is known as "ambient isotopy": 2 knot diagrams are ambient isotopic if they come from the same knot. It was shown in the 19th century that any two ambiently isotopic knot diagrams X and Y can be transformed into each other by the application of 3 alterations, known as Reidemeister Moves . These are:
Example: one can easily show through the application of the 3 Reidemeister moves, that the following configuration is equivalent to the Unknot:
The central problem of Knot Theory is to decide when two knot diagrams represent the same knot. This can be very difficult, and several classical polynomials, known as "knot invariants", have been developed to deal with large classes of diagrams: the Alexander and Alexander-Conway polynomials, the Jones polynomial, the Kauffman polynomial, the Homfly polynomial, and so on.
Two polynomials associated with different knots must be different. However, the same knot may be associated with different polynomials. To date, no-one has discovered a universal invariant that is unique for each knot. Let's put it this way: there is a way, via tensor calculus, to generate an invariant that is unique to each knot. It is about as useful to knot theorists as writing down the atomic structure of one's toes is for someone learning how to walk.
There are several ways by which the 3 stages of the bus boarding process can be represented through schematic diagrams resembling knots and braids . It would be in the interest of the Greyhound corporation, and perhaps other bus lines as well, to hire knot theorists to decide, from an examination of the knot, link and braid diagrams of all historical outcomes of stage C of the bus boarding process, which of them are generic in the sense that they are most often reproduced. With this information at hand Greyhound would then be in a position to issue insurance policies against injuries , like damaged luggage and broken limbs, arising from the vagaries of stage C.
One such scheme I have dubbed a "tangle box". It preserves the basic shape of the bus but provides no information on the temporal relations. Another, the "confrontation graph" , maps the long axis of the bus against time.
Our schematics are not adequate to deal with situations in which X actually drags Y down the length of the bus, or even ejects him altogether! This may be represented as follows:
Diagrams of this sort lie outside the purview of both knot and braid theory.
Although convinced that the Greyhound corporation ought to take a keen interest in my latest researches in this domain, I have not, despite many years of experience with these phenomena, proposed myself for the task of applying knot invariants to the uncovering of the generic configurations of stage C. The weakness of my geometrical intuition precludes my doing so.
One should not doubt that the brilliant modern school of young knot theorists will prove themselves more than equal to the task , as well as far more daunting challenges that must arise in the future.