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This text provides a guide to dealing with 3-manifolds by computers. Its emphasis is on presenting algorithms which are used for solving (in practice) the homeomorphism problem for the smallest of these objects. The key concept is the 3-gem, a special kind of edge-colored graph, which encodes the manifold via a ball complex. Passages between 3-gems and more standard presentations like Heegaard diagrams and surgery descriptions are provided. A catalogue of all closed orientable 3-manifolds induced by 3-gems up to 30 vertices is included. In order to help the classification, various invariants are presented, including the new quantum invariants.
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Series on Knots and Everything - Vol. 5
GEMS, COMPUTERS AND ATTRACTORS FOR 3-MAN I FOLDS Sostenes Lins
Gems, Computers and Attractors for 3-Manifolds
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. 6:
Knots and Applications edited by L H. Kauffman
Vol. 7:
Random Knotting and Linking edited by K. C. Milieu & D. W. Sumners
Vol. 9:
Combinatorial Physics T. Bastin & C. W. Kilmister
Vol. 10: Nonstandard Logics and Nonstandard Metrics in Physics W. M. Honig Forthcoming: Vol. 8: Symmetric Bends: How to Join Two Lengths of Cord R. E. Miles
E Series on Knots and Everything - Vol. 5
GEMS, COMPUTERS AND ATTRACTORS FOR 3-MANIFOLDS
Sostenes fins Departamento de Matematica Universidade Federal de Pernambuco Brazil
World Scientific Singapore •NewJersey•London* Hong Kong
Published by World Scientific Publishing Co. Pte. Ltd. P 0 Box 128, Farrer Road, Singapore 9128 USA office: Suite IB , 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data Lins, S6stenes. Gems, computers, and attractors for 3-manifolds / S6stenes Lins. p. cm. -- (K & E series on knots and everything ; vol. 5) ISBN 9810219075
1. Three-manifolds (Topology) -- Data processing . II. Series. QA613.2.L56 1995 514'.223--dc20
I. Title.
95-31016 CIP
British Library Cataloguing-in-Publication Data A catalogue record of this book is available from the British Library.
Copyright ®1995 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented without written permission from the Publisher.
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This book is printed on acid-free paper.
Printed in Singapore by Uto-Print
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Preface Crystallizations (and more generally, gems) have been around for almost two decades. Personally I rediscovered them in 1979. Since this time I have been playing with these objects. Gems (Graph encoded manifolds) encode closed manifolds by means of colored graphs. The prototype for realizing this encoding arises by taking the dual of the barycentric subdivision of a cell complex whose underlying topological space is a closed n-manifold. The 1-skeleton of such an object is an (n + 1)-regular graph whose edges can be properly colored with n + 1 colors. The construction is simple, but by itself has a serious drawback. The size of