Page 289 - J. C. Turner "History and Science of Knots"

280 History and Science of Knots

None of the encyclopedias described earlier give any ways of answering
such questions. Nor do any of the papers or books in the literature of topolog-
ical knot theory contain clues which might lead to answers. Indeed, it must be
said (and we shall elaborate on this below, using largely Schaake's own words)
that topological knot theorists are not at all interested in this kind of problem,
vital though it is in the real, physical world of braiding. A well-known remark
is that a topologist is a person who cannot distinguish between a doughnut
and a coffee-cup. Similarly, a topologist is a person who cannot distinguish
between a Reef Knot and a Lark's-head Knot. In the knotter's world, these
are different knots, with different properties and different ways of tying them.
In the early 1980s, Georg Schaake became aware of this gap, namely the
almost total lack*of general mathematical theories to model braiding processes;
and he decided to do something about it. Within a space of ten years he
discovered many beautiful relationships between numbers of parts (p), numbers
of bights (b) and numbers of strings ( s) in a braid; and these relationships
formed the basis of a broad general theory of braiding.
This theory is clearly a branch of number theory, with modular arithmetic
and solution of Diophantine equations under boundary conditions being main
elements of it. There is much appeal to polygonal grid diagrams, which are
precise drawings of the string-runs of braids; as a consequence , one can say that
Schaake's braiding theory is geometric and arithmetic; it is not topological.
It is a different kind of knot theory, different in goals and methods from the
mainline branch which has stemmed from Listing's and Gauss' work, and the
classification work of Tait, Kirkman and Little in the 80s and 90s of the last
century.
Since 1987 Schaake has been carrying out a project to publish his discover-
ies, collaborating with other authors to produce a stream of books, pamphlets,
articles and research papers on a variety of topics in braiding theory. Items
[15]-[24] of the bibliography of this Chapter indicate the extent of his pub-
lished output in a period of six or seven years. It ranges from the books
on Regular Knots, Fiador Knots and Herringbone Knots through to extensive
pamphlets on the braiding of Wheelknots. And it includes `spin-off' work in
Number Theory. For example, in 1987 a study of Schaake's evolution tree for
Regular Knots enabled Schaake to discover a new way of solving Pythagoras'
equation for integer triples (for sides of right triangles); all of them, including
their multiples, were given by this method, and they were classified in entirely
new ways, using tree graphs. In view of the ancient nature of this problem,
this was a remarkable feat. The work resulted in the book (with J. C. Turner
*One or two other authors have taken small steps to fill this gap in the past two decades;
for example, van de Griend [6] has written on Knots and Rope Problems and Karner [8]
has given tables for the development of Turk's Heads. But their work is insignificant beside
Schaake's.