Page 263 - J. C. Turner "History and Science of Knots"

254 History and Science of Knots

[73]; and Joan Birman has written a speculative and exciting overview article
[15] of the very latest developments.
After the discovery of the Alexander polynomial, knot literature came in
a steadily increasing flow. Nowadays one may speak of an explosive growth of
papers in the field. Yet comprehensive books, both monographs and textbooks,
are still few and far between. Kurt Reidemeister's pioneering work Knoten
Theorie of 1932 appeared in English translation [74] in 1983. Its approach,
of course, follows the combinatorial spirit of its times, and so only supplies
a historical introduction to the subject. Another book still of much value is
Introduction to Knot Theory, by Richard Crowell and Ralph Fox (1963) [25].
This gives a beautiful introduction to the subject from the classical algebraic
topological point of view, and is a fine tribute to the developments which
emerged in the post World War II period. Knots and Links by Dale Rolfsen
(1976) is remarkable for a number of reasons [77]. It is a giant leap into
(geometric) topology, and introduces all developments up to the mid-70s. An
excellent introduction to the theory of braids is Braids, Links and Mapping
Class Groups by Joan Birman [14].
Following the explosion of activity in applied knot studies in the late 80s,
a stream of books on the topic has been published. For example, Louis Kauff-
man's Knots and Physics [54], hard on the heels of books such as Braid Group,
Knot Theory and Statistical Mechanics (edrs. C. N. Yang and M. L. Ge, 1989)
and New Developments in the Theory of Knots (edr. Toshitake Kohno, 1990);
these last two are volumes 9 and 11 in World Scientific's Advanced Series in
Mathematical Physics. This present book is volume 11 in World Scientific's
Series on Knots and Everything. And in January 1992 the first edition of
Journal of Knot Theory and its Ramifications appeared, also published by
World Scientific; the subject has, at last, its own Journal.
It is inevitable that the new ideas and theories about knots will gradually
be introduced into syllabuses for graduate and undergraduate mathematicians
and physicists. Textbooks for teaching the subject will come forth. An ex-
cellent recent example is Knot Theory, by Charles Livingston (Mathematical
Association of America, 1993); he covers much of the classical theory, and con-
tinues through to high-dimensional knots and the combinatorial techniques of
various of the new polynomial invariants. He includes many exercises suitable
for undergraduates, to whet their appetites and help them come to grips with
this exciting but demanding subject.

References

1. Y. Akutsu and M. Wadati, `Knots, Links, Braids and Exactly Solv-
able Models in Statistical Mechanics', Communications in Mathemati-
cal Physics, 117, (1988). 243-259.