Page 290 - J. C. Turner "History and Science of Knots"

On Theories of Knots 281

[24]) entitled A New Chapter for Pythagorean Triples, and a research report
[22] on new methods for solving quadratic Diophantine equations.
This publishing project is continuing. Schaake has tackled many more
problems and knot classes than appear in the works Referenced in this book.
His further results are steadily being written up and published in one form or
another.
We shall end this essay by giving brief notes on the philosophy, basic
ingredients and methods of Schaake's braiding theory. The final remarks will
draw heavily upon Schaake's own statements, made in Pamphlet No. 8, where
he gives his views upon Knots-Facts and Fallacies.

6. Basic Ingredients and Philosophy of Schaake 's Theory of Braiding

In order to begin studying Schaake's theories, it is necessary to know what is
meant by the following terms: parts, bights, string runs, coding, grid diagram,
algorithm table. They are used in reference to a braid*which can be constructed
by passing a string from left-to-right, right-to-left, and so on, round and around
a cylinder, interweaving the string with itself by making a succession of under-
and over-crossings as the passes are made.
The `grid diagram' is a geometric picture which represents the braid (and
the braiding process) completely. The `algorithm' gives a list of instructions for
carrying out the interweaving as the string passes round and around the cylin-
der. The `coding' is the particular pattern of string crossings which appears on
the grid diagram. A `string run' is a rectilineal path which the `string' takes as
it traverses the grid diagram, from the left boundary to the right boundary or
vice versa. Such grid diagrams are best drawn on isometric grid graph paper.
We shall illustrate these terms and ideas in relation to the production of the
very simple and familiar 3-crossing knot, namely the trefoil knot.

6.1. Diagrams and Production Steps for the Trefoil Braid
The diagrams (a) and (b) in Fig. 5 are well-known representations of the right-
handed `trefoil knot' and braid respectively. Note that (a) is formed from a
single closed string; whereas (b) is formed from two interwoven strings, one
running from point P1 to Q2, and the other from P2 to Q1.
Next note that if a circular cylinder be placed behind the braid (b) with
its straight-line generators parallel to P1 P2 and Q1Q2i then the braid can be
wrapped backwards around the cylinder until P1 meets Ql and P2 meets Q2.
Let us assume this done, and that we have joined the ends P1 and Q1,

*`No distinction is to be drawn between `braid' and `knot', since any knot is in fact a braided
structure.'