Page 297 - J. C. Turner - History and Science of Knots

On Theories of Knots 287

Georg Schaake has discovered the methods necessary to obtain algorithms
for braiding of knots in many classes-for different sets of conditions on their
string runs and codings. It is not possible to give meaningful summaries, in
this short chapter, of any of the formulae or mathematical methods involved
in this work. The list of his achievements is long. To name a few of the classes
he has dealt with, we begin with the general class called Regular Knots, which
includes many well-known types of Knot such as Turk's Head, Head Hunter's,
Gaucho, Bot6n Oriental, Fan, Ring, Slow and Fast Helix and Basket Weave-
all knots well-known to experienced braiders. The list of known knots in
this class could go on. We emphasise that he has dealt with these classes (or
subclasses) in great generality; in principle, from his formulae braiding algo-
rithms can be computed for any given numbers of crossings, for each of the
named types. Indeed, his collaborators have produced computer programmes
from his formulae, which produce the required algorithms whenever particu-
lar parameter values are input. In the Knot Encyclopedias described earlier,
just special cases of these braids were treated, with drawings and graphical
algorithms given for each. It was descriptive work only.
To continue the list of his achievements, he has studied many general types
of Regular Knots-for example those with Row-Coding, with Column-Coding,
or with Casa-Coding (see [15, Pamphlets 6, 7, 9, 10]).
Having dealt with single-string braids, one can turn one's attention to
multi-string braids. The possibilities for extending the types and complexi-
ties of braids for making, studying and classifying grow exponentially. For
example, if one relaxes the condition that all the bights of the braid must lie
on two parallel edges, the possibilities become truly vast. Of course, Schaake
first turned his attention to those knots which are of most interest to artisan
braiders-and the types of knot which they wish to braid are generally both
functional and decorative (that is, artistically pleasing). Beauty is their guide,
when they select coding patterns for their braided objects. If further consid-
erations are added, such as colours of strings (they can be all the same, or be
selected from a range), then again, the numbers of possible braid-types within
each weave-type class goes up rapidly.
Some of these more complex types, the problems of which Schaake has
solved in broad generality, include the Standard Herringbone Knots, the Reg-
ular Fiador Knots, the Standard Herringbone Knots, and Wheelknots. The
last-named Knots are different from all the earlier, cylindrical type braids, in
that they involve braiding in the form of a torus; moreover, holes have to be left
in the weaving, at regular intervals around the torus, for wheel-spokes to pass
through. To imagine this, think of a steering wheel of a yacht, with perhaps
five spokes; the braiding process, using one string only, has to pass around the
circumferential rim of the wheel, constructing the required coding pattern as
it goes and avoiding the spokes neatly. In [15, Pamphlet 12] Schaake gives