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the world. Alexander, frustrated by the difficulty of the knot, took his sword
and cut it asunder. Perhaps this story can be adapted for those who are
engaged in developing the newly emerging knot science. The problems posed
by knots are legion; and the rewards for solving them are potentially great.
Solvers will surely fall under the spell of the magic of knots. And they will
meet with many setbacks and frustrations. However, they are unlikely to find
sharp. solutions, like Alexander's sword: they must forge ever more powerful
mathematical tools to overcome their difficulties.

Acknowledgements
We wish to thank all those who have helped to bring this book into being. Our
special thanks go to Charles Warner, for the great help he gave in the design of
the book, and for his careful editing of many of the chapters. Heartfelt thanks
are due to Bill Rogers, of the University of Waikato, New Zealand, who has
given much help with the computer type-setting; without his advice, and his
ever ready willingness to solve our I4TEX computing problems, it would have
taken us very much longer to put the book into camera-ready form.
John Turner gives warm thanks to his colleague Georg Schaake, for the
very many pleasant hours spent together discussing Schaake's theory of braid-
ing, and working on publishing and research projects. He acknowledges the use
in Chapter 12 of many diagrams and quotations from the books and pamphlets
of Schaake, Turner et al.
With regard to copyright questions, every author was asked to take care to
acknowledge any quotations used, making adequate references to their sources.
If they used diagrams from research journals or books, they were asked to
write to the copyright holders for permission to do so, if deemed necessary. It
is evident that in a book such as this, all of our authors will have drawn on
very many sources for much of their information. We wish to acknowledge,
collectively, our debt to all these sources.
Finally, we thank the twelve authors who between them produced the
eighteen essays which comprise this book. We thank them for their scholarship,
their authorship and their patient help in bringing the essays to their final
chapter forms.
The authors submitted biographical notes on their lives and work, and
these are gathered together in a section entitled About the Authors. It may
be found at the end of the book.

Pieter van de Griend
John Turner
1996