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Seiichi Kamada

Surface-Knots in 4-Space An Introduction

Springer Monographs in Mathematics Editors-in-Chief Isabelle Gallagher, Paris, France Minhyong Kim, Oxford, UK

More information about this series at http://www.springer.com/series/3733

Seiichi Kamada

Surface-Knots in 4-Space An Introduction

123

Seiichi Kamada Graduate School of Science Osaka City University Osaka Japan

ISSN 1439-7382 ISSN 2196-9922 (electronic) Springer Monographs in Mathematics ISBN 978-981-10-4090-0 ISBN 978-981-10-4091-7 (eBook) DOI 10.1007/978-981-10-4091-7 Library of Congress Control Number: 2017933445 © Springer Nature Singapore Pte Ltd. 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Dedicated to Naoko

Preface

Knot theory is one of the most active research fields in modern mathematics. Knots and links are closed curves (1-dimensional manifolds) in the Euclidean 3-space, and they are related to braids and 3-manifolds. These notions are generalized into higher dimensions. Surface-knots and surface-links are closed surfaces (2-dimensional manifolds) in the Euclidean 4-space, and they are related to 2-dimensional braids and 4-manifolds. Surface-knot theory treats not only closed surfaces but also surfaces with boundaries in 4-manifolds. For example, knot concordance and knot cobordism, that are also important objects in knot theory, are surfaces in the product space of the 3-sphere and the interval. Although the beginning of the study of surface-knots is due to E. Artin in the 1920s, the crucial research was started by R.H. Fox using the motion picture method, in the 1960s, followed by J. Milnor including researches on knot conc

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