Page 301 - J. C. Turner "History and Science of Knots"

292 History and Science of Knots

remarks, together with other of his views on knots, drawn from that pamphlet.
As with the Figures above, we shall pass them on without further comment.
They illustrate his strong beliefs, based on his fifteen years work with his own
knot theory, as to what should be the main thrust of knot research.
`This Pamphlet is mainly intended for the general reader interested in
knots and braids. It is also intended for those who have always wondered why
the academic world has not progressed beyond their "classical knot theory"
and consequently has never yet been able to produce a sound knot theory of
practical value ... '

`It is regrettable that any association which their emerging "knot theory"
might have had in the past with physical knots, has been lost without recog-
nizing this.' [Schaake is here referring to the fact that topological knot theory
does not treat of physical aspects and properties of knots.]

`The "classical knot theory" is concerned with imaginary "closed" knots
only, and hence it is not able to address the vast majority of real knots.'

`We have shown above that the "classical knot theory" is a theory of
hypothetical knots and braids, hence a theory which has no bearing on real
knots. It is therefore not surprising that it cannot give constructional directives
for the creation of knots and braids. By its very nature it is a theory in a
static framework (only "closed" hypothetical knots are studied). Knots and
braids are dynamic objects, resulting from creative (evolutionary) processes,
and hence require a theory in a dynamic framework. Therefore, it should
be obvious that the required theory should concern itself with the question:
How does a real knot or braid come forth?'

`We have already mentioned that knots and braids are in essence of a
geometric nature. Hence it is of vital importance not to lose their geometrical
characteristics in any modelling processes. The "classical knot theory" how-
ever, disregards the geometrical aspects, and treats knots in a purely topolog-
ical manner. This leads to various results which are ambiguous when applied
to physical, hence real, knots and braids.'

`We mentioned that in the "classical knot theory" all knots are closed
structures, and hence before studying a knot it gets closed. We have already
seen that a "closed" Overhand Half Knot and a "closed" Overhand Knot both
result in a Right-Hand Trefoil. Now let's again close these two knots after
everting* and investigate the result. In the case of a "closed" everted Overhand

-everting' means `turning inside out'; here he is referring to cylindrical braids, which have