Page 262 - J. C. Turner "History and Science of Knots"

A History of Topological Knot Theory 253

From the point of view of contemporary topological knot theory, the chief
problem is to find an interpretation of the new invariants in terms of classical
algebraic topology (homology theory, homotopy theory and such) or differen-
tial geometry (differential forms, connections, etc.). Here considerable specu-
lation has produced little of note. Perhaps there can be no such interpretation,
and states-model theories from statistical mechanics must be incorporated into
topology. Perhaps we shall witness the emergence of other, exciting and en-
tirely new theories, such as quantum mathematics, enriching mathematics.
Who can tell? The recent interactions between knot theory and the rest of
mathematics are really quite bewildering. They indicate that there is much
still to be done.
Knot theory as it stands today represents a significant stream of ideas,
flowing from the challenging difficulties of describing and understanding the
phenomenon knot and its observable properties. The abstract heights it has
reached, and the applications it has so surprisingly found in the wake of Jones'
discoveries, give eloquent support to the often-mentioned notion of: `the seem-
ingly inevitable utility of mathematics conceived symbolically without reference
to the real world.'
It has been said that knots are more numerous than the stars, and are
equally mysterious and beautiful. Like the stars seen at night, knots pervade
our senses and challenge us to understand them. This happens now, not only in
our everyday working world but also, as we learn from the quantum physicists,
in our deeper philosophical efforts to explain the mysteries of fundamental
physical and biological phenomena. The needs to understand these mysteries
will continue to give impetus to the currently widening spread of research into
theories and applications of knot theory.

Bibliographic Notes

Knot theory has a substantial literature, albeit very scattered; literature on
the history of the subject is also scattered, fragmentary and sporadic. The
earliest works, before the turn of this century, tend to mention many interest-
ing sources; but as a rule authors on knot theory after 1900 are rather sparing
with their historical information. Luckily there are a few exceptions such as
Dehn/Heegaard [29]. From a mathematician's point of view, undoubtedly the
most impressive accounts of knot theory's history may be found in Gordon
[39] and Thistlethwaite [86]. The encyclopaedic work by Burde/Zieschang [22]
evaluates and records the state of the field immediately before the spectacular
discovery of the Jones polynomial. Their book supplies fragmentary histori-
cal data; but their bibliographic listing has over 1000 entries to compensate.
Wilhelm Magnus has written about the early history of braid theory in [64].
Jozef Przytycki has described parts of the modern history of knot theory in