Table Of ContentHISTORY AND
SCIENCE OF KNOTS
SERIES ON KNOTS AND EVERYTHING
Editor-in-charge: Louis H. Kauffman
Published:
Vol. 1: Knots and Physics
L. H. Kauffman
Vol. 2: How Surfaces Intersect in Space
J. S. Carter
Vol. 3: Quantum Topology
edited by L. H. Kauffman & R. A. Baadhio
Vol. 4: Gauge Fields, Knots and Gravity
J. Baez &J. P. Muniain
Vol. 5: Gems, Computers and Attractors for 3-Manifolds
S. Lins
Vol. 6: Knots and Applications
edited by L. H. Kauffman
Vol. 7: Random Knotting and Linking
edited by K. C. Millets &D. W. Sumners
Vol. 8: Symmetric Bends: How to Join Two Lengths of Cord
R. E. Miles
Vol. 9: Combinatorial Physics
T. Bastin & C. W. Kilmister
Vol. 10: Nonstandard Logics and Nonstandard Metrics in Physics
W. M. Honig
Vol. 11: History and Science of Knots
edited by J. C. Turner & P. van de Griend
Vol. 12: Relativistic Reality: A Modem View
J. D. Edmonds, Jr.
Vol. 13: Entropic Spacetime Theory
J. Armel
Vol. 14: Diamond - A Paradox Logic
N. S. K. Hellerstein
Vol. 15: Lectures at Knots '96
edited by S. Suzuki
Vol. 16: Delta - A Paradox Logic
N. S. K. Hellerstein
Series on Knots and Everything - Vol. 11
H ISTORY AND
SCIENCE OF KNOTS
editors
J C Turner
Universityof Waikato, New Zealand
P van de Griend
Aarhus University, Denmark
World Scientific
Singapore •NewJersey•London -Hong Kong
Published by
World Scientific Publishing Co. Pte. Ltd.
P 0 Box 128, Farrer Road, Singapore 912805
USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661
UK office: 57 SheltonStreet, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
History and science of knots / editors, J. C. Turner, P. van de Grien4.
p. cm. -- (K. &E serieson knots and everything: vol. 11)
Includes index.
ISBN 9810224699
1. Knots and splices-- History. 2. Knot theory. I. Turner, J. C.
(John Christopher), 1928- . II. Griend, P. C. van de.
III. Series.
VM533.H57 1995
623.88'82--dc2O 95-42872
CIP
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
First published 1996
First reprint 1998
Copyright m 1996 by WorldScientific PublishingCo. Pte. Ltd.
All rights reserved . Thisbook, or parts thereof may not be reproducedinanyform or byanymeans,
electronic or mechanical,including photocopying,recording or any information storage and retrieval
system now knownor to be invented,without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright
Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to
photocopy is not required from the publisher.
Printed in Singapore.
CONTENTS
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Part I. PREHISTORY AND ANTIQUITY
1. Pleistocene Knotting . . . . . . . . . . . . . . . . . . . . . 3
2. Why Knot?-Some Speculations on the Earliest Knots . . . . . . 19
3. On Knots and Swamps-Knots in European Prehistory . . . . . . 31
4. Ancient Egyptian Rope and Knots . . . . . . . . . . . . . . . 43
Part II. NON-EUROPEAN TRADITIONS
5. The Peruvian Quipu . . . . . . . . . . . . . . . . . . . . . 71
6. The Art of Chinese Knotwork: a Short History . . . . . . . . . . 89
7. Inuit Knots . . . . . . . . . . . . . . . . . . . . . . . . 107
Part III. WORKING KNOTS
8. Knots at Sea . . . . . . . . . . . . . . . . . . . . . . 135
9. A History of Life Support Knots . . . . . . . . . . . . . . . 149
Part IV. TOWARDS A SCIENCE OF KNOTS?
10. Studies on the Behaviour of Knots . . . . . . . . . . . . . . 181
11. A History of Topological Knot Theory . . . . . . . . . . . . 205
12. On Theories of Knots . . . . . . . . . . . . . . . . . . . 261
13. Trambles . . . . . . . . . . . . . . . . . . . . . . . . . 299
14. Crochet Work-History and Computer Applications . . . . . . 317
vi History and Science of Knots
Part V. DECORATIVE KNOTS AND OTHER ASPECTS
15. The History of Macrame . . . . . . . . . . . . . . . . . . 335
16. A History of Lace . . . . . . . . . . . . . . . . . . . . . 347
17. Heraldic Knots . . . . . . . . . . . . . . . . . . . . . . 381
18. On the True Love Knot . . . . . . . . . . . . . . . . . . 397
About the Authors . . . . . . . . . . . . . . . . . . . . . . 419
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
PREFACE
Knots are more numerous than the
stars; and equally mysterious and
beautiful ... John Turner (1988)
The importance of twines, cords, strings and ropes in the lives of humankind
is immense. All civilizations have depended upon them for their safety, and
for many other activities and uses which have contributed towards their well-
being. These activities range from house, bridge and boat building to weaving
and cloth production; from construction of fishing knots and nets to mainte-
nance of record-keeping systems; from making of articles of apparel to deco-
rative braiding of bags, belts and wall-hangings.
For much of mankind's history, knots and knot lore were closely linked
with magic, medicine and religious beliefs. Knots have even served as bases for
mathematical systems (by the Mayans, for example), before writing skills were
available; and of course, string games and other recreational uses were and are
legion. All of these activities have been practised since time immemorial, in one
form or another, and by all races. They are still practised today, throughout
the world. Furthermore, it is safe to say that they will go on being practised
until the day arrives that humankind no longer exists.
The original intention for this book was that a collection of essays should
be gathered, to form a useful, entertaining and authoritative account of the
history of many of these activities. We drew up a list of topics that might
be included-anything that involved a `crossing of strings' was considered-
and we came to realise that the whole field of knots and knotting applications
simply could not be covered properly in a single book. To do the subject full
justice an encyclopaedia would be needed. We had to be selective.
The Contents page shows the topics that we finally were able to include.
Notable and regrettable omissions from the list are ropemaking and rope tools,
basketry, weaving, knitting, tools for knotting, and uses of knots in magic,
medicine and religion. Much more could have been included, too, on the special
uses of knots made by many of the great civilizations that have flourished in
world history. We were able to include a chapter on Chinese knots; but we are
silent on the knotting practices of the Japanese, the Indians, the Arabs, the
Babylonians, and so on. We knew no authorities on knots of these races.
vii
viii History and Science of Knots
Nevertheless, the eighteen essays in the book cover a very wide range
of knot topics, and each deals with historical aspects of its own topic. The
remote history of our subject is generally shrouded in mystery and mythology;
for example, the art of making fish nets is said to have begun with Aphrodite
(a Greek goddess of love and beauty), who rose from the sea each morning and
gathered women around her to teach them how to make nets and to weave
and spin*. Our book begins with a search for less imaginative explanations of
how mankind began to make and use knots. Our desire has been to establish
the subject as much as possible on a scientific base. In truth, real evidence
on which to base the story of knots only arises when archaeological finds are
made, of artifacts which either still have rope and knots attached to them,
or which can reasonably be said to have been associated with use of rope,
twine or knots. Our first four Chapters deal with matters of this kind; many
archaeological discoveries pertaining to knots are described, and speculations
made about them.
Other sources of knot-history are the occasional references to the subject
matter which have been found in early writings of the ancients; and knowledge
gleaned from wall paintings in caves, heiroglyphs (for example, see Fig. 1), clay
tablets, papyri, quipus, tomb finds, pottery, carvings, and so on. However, it
has been noted that clear depictions of knots are extremely rare on ancient
paintings or earthenware; a reason mooted for this is that knots bore magical
or religious significance, and artists were afraid to portray them too faithfully.
A& I
Fig. 1. The ancient Egyptian hieroglyph for the word `tjeset' = knot
Indeed, the oldest written source on the subject (containing the only known
descriptions of knots before the 18th century) is the Iatrikon Synagogus, a
medical treatise compiled by a Greek physician, Oribasius of Pergamum, in
the 4th century A.D. He states that he derives his information from Heraklas,
who is believed to have been a physician who flourished about A.D. 100. The
knots recorded in this book were those used by physicians for slings, or as
parts of slings, during operations or in the treatment of bone fractures. They
are carefully described by C. L. Day in The Art of Knotting and Splicing (U.S.
Naval Institute, Annapolis, Maryland, 1970).
*In Norse mythology it is said to have been Loki, the god of mischief and destruction, who
taught mankind the art of netting.
Preface ix
It seems that it was not until the late 17th century and just beyond,
when mathematics flourished and scientific reasoning began to lead the minds
of great thinkers, that knots became worthy elements of study, in searches for
rational nautical knowledge. In the mid-18th century the grand encyclope-
dia of Diderot and D'Alembert appeared, and knots and their uses became
recorded as part of man's heritage. It was about this time that shipbuilding
was becoming more than an art: methodical thinking was being given to hull
shapes, strengths of spars, and ropes and riggings. Studies of principles of sea-
manship began to be made too. By the end of the 18th century, books such
as Hutchinson's A Treatise on Practical Seamanship, with Hints and Remarks
Relating Hereto: designed to contribute something towards fixing rules upon
philosophical and rational principle (1777), and Steel's Elements and Practice
of Rigging and Seamanship (1794), had appeared. Textbooks bearing instruc-
tion on seamen's knots found ready sale, as the need for trained ship's officers
grew rapidly with the great expansion of the British Navy during the period of
the Napoleonic Wars. As is so often observed in history, advances of a science
are greatest in times of war.
In view of the venerable and extensive use of knots over the long mil-
lennia, it is somewhat surprising that they did not become a topic deemed
worthy of mathematical study until the late eighteenth century. No doubt
there had been many attempts in the past to discover the most efficient kinds
of knot, and the best kinds of rope to use, in specific applications-for exam-
ple in seamanship. But it seems that before about A.D. 1771 no-one saw any
possibility* of modelling knots mathematically; one certainly couldn't apply
Euclidean geometry to do so-and that generally was firmly believed to be
the mathematics of God's plan for the Universe and all that it contained.
It took the genius of a man like Gauss, one of the greatest mathematicians
and astronomers the world has known, to break this taboo. In 1794 he prepared
sketches of thirteen knots (Fig. 2), with English names, perhaps copied from an
English book of seafarers' knots; this note, together with other papers bearing
sketches of knots, was found amongst his papers after his death in 1855. It is
cleat from his notes that he considered the study of knots to be an important
task for mathematicians to attempt; and he encouraged some of his students
to engage in that study. He himself published only one paper in the field.
The story of these beginnings of topological knot theory, and of its gradual
development into what it has now become, a major branch of Topology, is told
in Part IV, Chapter 11.
*In 1771, A. T. Vandermonde (1735-1796) published a paper in Memoires de 1'Academie
Royale des Sciences, Paris, in which he suggests that `the craftsman who fashions a braid,
a net, or some knots will be concerned, not with questions of measurement, but with those
of position: what he sees there is the manner in which the threads are interlaced.' The title
of his paper was `Remarques sur les problemes de situation.'
x History and Science of Knots
G/3^ /yiaillr 33
Fig. 2. Copy of a page from the notebooks of C. F. Gauss (1777-1855)
There has been considerable discussion amongst authors and editors of
this book as to whether the term `Science' can legitimately be applied to
describe the study of knots. As Professor C. E. M. Joad would have said,
when speaking on the British radio programme Brains Trust, during World
War II: "It all depends on what you mean by `Science'." We have reached
no agreement! Chapter 12 discusses this issue, and also outlines some non-
topological knot theories that have emerged recently.
We have organised the eighteen chapters into five sets, or Parts, to impose
what we believe to be a useful ordering upon them.
