Page 217 - J. C. Turner - History and Science of Knots

A History of Topological Knot Theory 207

a proper mathematical discipline, in the second half of the 19th century, knot
theory has been associated with many projects on the frontiers of fundamental
scientific research. Although much of knot theory is mathematically abstruse,
it has found important application in fields as diverse as atomic modelling,
quantum physics, theoretical psychology [82] and molecular biology [95].
The table of (Fig. 1) gives an approximate chronology of the chief dis-
coverers and developers of Knot Theory. The column to the far left indicates
the time scale, in years. The second column gives some of the great names in
topology. The split column to the right is used to indicate the historic placings
of the greatest names in Knot Theory, and also to symbolise its dual nature:
the left branch covers mathematical knot theory, whilst the right branch caters
for the mathematical theory of braids.

2. The Very Early Days
Throughout history many different interpretations of the phenomenon knot
have been proposed. In this section we shall consider knots to be those struc-
tures which can be realized in a knottable medium, such as a length of rope.
In order to show how modelling of the simple act of tying a knot progressively
translates into amazingly complex mathematical machinery, we shall first be
concerned with questions about the awareness of mathematical problems as-
sociated with knots. In this context prehistory is defined to be the interval of
time before the recording of data or information in formalized mathematical
ways began to take place.
Although knot theory exists nowadays as a highly specialized and concrete
set of mathematical ideas, its origins are not easily traced. Yet somewhere in
a far and distant past, inklings of those ideas must have been born. Unfor-
tunately, in a mathematical perspective, the theory's tremendously dispersed
history prevents them from being localized with certainty. One may be in-
clined to believe that there was no theorizing about knots in the very early
days of Mankind. Surely there was none in any of the stringent ways we now
use to model the phenomenon knot. Nevertheless, by making certain assump-
tions, plausibility for which is provided by results from anthropological and
psychological studies concerning knots [26], [82], one is able to make state-
ments about the roots of the theory. Hence some tracking of the subject's
gradual evolution is feasible.
The discovery and use of knots would seem to predate those of fire and
the wheel by countless aeons. Knots were used long before the practice of
mathematization began to influence the thoughts and actions of Mankind. To
primitive Man even the simplest of knots would pose vexing and crucial prob-
lems. Yet, it is doubtful whether he would analyze them in any degree other
than what was required to employ them as practical tools in his struggle for