Page 287 - J. C. Turner "History and Science of Knots"

278 History and Science of Knots

knot. Schaake gives 112 grid diagrams* of 4-crossing knot forms! The above
Figure (see Fig. 4) shows 16 of these. It will be noted that they are open-
ended forms, with S, W referring respectively to the `standing-end' and the
`working-end' of the knot string. Schaake defines five different equivalence
relations on these forms, namely Lateral, Evert, Evert-Lateral, Transpose,
and Convert, and discusses classifications of the 112 forms in terms of these
relations. We cannot go into details of these relations here, for space reasons.
We shall merely quote one of Schaake's comments, with regard to Convert
string manipulations, which points up fundamental differences between his
kind of knot theory and that of the topologists:

`We shall see later, with the aid of examples, that with string ma-
nipulations associated with Convert equivalency , the results for
`closed' knots are generally quite different from those obtained for
open-ended knots of similar structure. This is, of course, not sur-
prising, since the result obtained with the described string manip-
ulations (in finite space) on a knot `closed' in finite space is not
the same as that with the same knot closed at infinity (which is
equivalent to the continuation of the string-ends to infinity).'

Having said a little of his methods, with reference to the 4-crossing knots, let
us return to our description of the man and his work.
Georg Schaake is a superlative craftsman in many fields, from bone carv-
ing to glass engraving, from leather braiding to decorative knotting. In all
his endeavours, he brings an original mind to the task of understanding their
principles and improving their practices.
He has been particularly successful in developing mathematical theories
which describe braiding processes. When he took up leather braiding in the
early 80s, and read the literature on the subject, he was immediately struck by
the fact that there were no satisfactory definitions of knots, and no mathemat-
ical rules by which they could be tied and studied further. He wondered why
the whole treatment of the formation of knots had not been placed (and long
since) on a surer footing than the graphical, wholly empirical ones given in
the encyclopedias of Ashley [1] or Grant [5]. Their works are monumental, of
great beauty, worth and scholarship; but they espouse no systematic method
for producing knots and braids, no theoretical treatment.
Among the very few references to mathematics in these works, the fol-
lowing two quotations are revealing. The first (from [5, p. 440]) occurs within
a two-page section headed `The Braiding Detective'. It gives the only clues
to the reader that in the wonderful and infinitely extensive world of braiding

*The `grid diagram' is Schaake's basic tool for displaying and analysing knots; through its
use, his theory becomes both geometric and analytic.