Page 288 - J. C. Turner "History and Science of Knots"

On Theories of Knots 279

there lies the possibility of a theory of braiding, and of methods by which the
artisan can develop and analyze his own creations of decorative braids.
`Duplicating Strange Braids and Creating New Ones. The
braided knot, or even the Turk's-head, is a mathematical marvel.
It goes round and round and comes out perfect-the working end
of a braided knot returns to its point of origin.
It is interesting to analyze, more absorbing to create, and definitely
a challenge. Distinct variations are limitless and all of them are
adjusted to a mathematical scale of easy deduction.

Dr Almanzor Marrero y Galindez, in the prologue of his book, Cro-
mohipologia, said of his Argentine father: `During his last years
almost each day he made an original criollo button, which he first
posed as a problem, resolving it mathematically, and later exe-
cuting ther formulae he had worked with success. Some of these
buttons were of such complexity that he had to use several strings
in the same knot, whose total length reached several metres.'
Alas, his mathematical techniques in planning his buttons have
been lost to posterity. Many serious scientists and mathematicians
have given profound thought to the Turk's-head, and as an inde-
pendent result of the studies of Clifford W. Ashley of New England,
and George H. Taber of Pittsburgh, the `Law of the Common Di-
visor' was discovered.'
The Law of the Common Divisor for single-string Turk's-head knots, as enun-
ciated by Ashley in [1, p. 2331, is as follows:
`A knot of one string is impossible in which the number of parts
and the number of bights have a common divisor [greater than 1].'
This would seem to be the extent of any known theory of braiding processes,
before Georg Schaake began his studies. Moreover, it seems that the above
law was arrived at empirically; no proof was given for its general truth. We
were unable to trace any further of Ashley's or Taber's writings on the matter.
Schaake and Turner give a concise proof of the Law in [20], and a discursive
one in [26].
There appear to be no extant mathematical theories, certainly no general
ones, before Schaake's, which enable algorithms (step-by-step methods) to
be calculated and written down so that artisans can produce braids of given
patterns. Without such a theory, of course, fundamental questions about the
universe of braids cannot even begin to be answered. For example, we might
ask: `Suppose we wish to produce a braid with a certain herringbone pattern
of string-crossings, using only one string and having 95 parts and 36 bights.
Can it be done? If so, how may it be done? Give us a method to do it.'