Page 295 - J. C. Turner "History and Science of Knots"

286 History and Science of Knots

P = (P1, p2, ... , pn) of parameters. The question can be asked `Which integer
vectors will correspond to knots in that class?'
It is the pursuit of answers to this kind of question which has caused the
greatest fascination and pleasure to this author. The solutions often provide an
evolution tree, which can take the form of a number tree[25] which consists of
a tree graph displaying the parameter vectors on its nodes; the paths between
nodes indicate relationships between knots in the given class. It takes little
imagination to view these trees as new objects in number theory. Every study
of their properties constitutes (so this author claims) new investigations into
properties of numbers. To give just one example of the charm and power of
such studies, we have already mentioned how, in 1987, we discovered new ways
for solving second-order Diophantine equations [22], by studying the Regular
Knot evolution tree.
The production of a step-by-step plant for obtaining a braid, having a
given string run and coding pattern, is traditionally done by tracing the braid
pattern on paper and following it around carefully from a starting point, noting
down the crossing-types as they are to occur along the string.
Exactly the same procedure can be followed, though often very tediously,
using Schaake's grid diagram for the desired braid. The algorithm for produc-
ing the braid by passing string back and forth over a cylinder can be similarly
determined. We showed how to do this in the simple trefoil example, given
above, following the string run and observing the interweaving that had to
take place within each half-cycle.
In principle, this `pencil-and-paper' approach can always be used; even
when the braid has many hundreds of crossing-points. However, it is desirable
to have mathematical methods to compute the algorithm in more complex
cases. Such mathematical methods not only provide the algorithm but also
deepen one's understanding of the processes involved; and they can be gener-
alized.
This might be said to be the point where Schaake's theory of braiding
becomes a mathematical science. The grid diagrams of braids (any braid can
be represented by one) are the basic mathematical objects of that science.
They are objects which are essentially geometric (although some topological
deformations can be made on a diagram, without destroying or corrupting the
information it carries about its braid). The types of mathematics which are
used to analyse grid diagrams are those of geometry, modular arithmetic, and
systems of recurrence equations with many boundary conditions.
*An easy example is the following. In the class of Regular Knots, the whole string run of
a knot is determined by an integer pair (p, b). It is easy to show that any ordered pair
(m, n) of whole numbers corresponds uniquely to a Regular Knot string run provided that
g.c.d.(m, n) = 1.
Compare the use of a knitting pattern to produce a jersey.