Braid and Knot Theory in Dimension Four
Braid and Knot Theory in Dimension Four
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DOI：
10.1090/surv/095
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发表时间：
2002-05
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通讯作者：
S. Kamada
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作者：
S. Kamada
辫子理论和纽结理论通过亚历山大和马尔可夫的两个著名结果联系起来。亚历山大定理指出，任何结或链接可以放入辫子的形式。马尔可夫定理给出了两个辫子代表同一个结或链接的必要和充分条件。因此，人们可以用辫理论来研究纽结理论，反之亦然。在本书中，我们将辫子理论推广到四维。我们发展了表面辫的理论，并将其应用于研究表面链接。特别地，给出了四维情形下的广义亚历山大定理和马尔可夫定理.这本书是第一个地方，包含了一个完整的证明广义马尔可夫定理。本文还利用运动图像法研究了曲面连接，并对运动图像法中的一些关键技术进行了研究。对于表面编织，介绍和发展了各种描述方法：运动图像法、图表描述法、编织单值法和编织系统法。这些工具是理解和计算表面辫和表面链接的不变量的基础。一个表的纽结表面包括与计算的亚历山大多项式。辫子技术扩展到表示链接同伦类。美国国会图书馆编目在出版D a t a Kamada，Seiichi，1964在第四维度的编织和结理论/ Seiichi Kamada. p.cm。- （数学调查和专著;五95）包括参考书目和索引。ISBN 0-8218-2969-6（alk.纸）1.编织理论。2.纽结理论。I.标题.二.数学概论和专著;第95期。QA612.23.K36 2002 514.224-dc 21 2002018274修订和重印。本出版物的个人读者和代表他们的非营利图书馆被允许合理使用材料，例如复制一个章节用于教学或研究。允许在评论中引用本出版物中的简短段落，前提是通常承认来源。再版、系统复制或多次复制本出版物中的任何材料仅在美国数学学会许可下允许。这种许可的请求应向采购部，美国数学学会，邮政编码。信箱6248，普罗维登斯，罗得岛02940-6248。也可以通过电子邮件向美国数学学会的reprint-permissionOams.org. © 2002提出请求。All rights reserved.美国数学学会保留除授予美国政府的权利外的所有权利。在美利坚合众国印刷。@本书所用的纸张是无酸的，福尔斯符合为确保持久性和耐用性而制定的准则。访问AMS主页，网址：http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 07 06 05 04 03 02
Braid theory and knot theory are related to each other via two famous results due to Alexander and Markov. Alexander's theorem states that any knot or link can be put into braid form. Markov's theorem gives necessary and sufficient conditions to conclude that two braids represent the same knot or link. Thus one can use braid theory to study knot theory, and vice versa. In this book we generalize braid theory to dimension four. We develop the theory of surface braids and apply it to study surface links. Especially, the generalized Alexander and Markov theorems in dimension four are given. This book is the first place that contains a complete proof of the generalized Markov theorem. Surface links are also studied via the motion picture method, and some important techniques of this method are studied. For surface braids, various methods to describe them are introduced and developed: the motion picture method, the chart description, the braid monodromy, and the braid system. These tools are fundamental to understanding and computing invariants of surface braids and surface links. A table of knotted surfaces is included with a computation of Alexander polynomials. The braid techniques are extended to represent link homotopy classes. Library of Congress Cataloging-in-Publication D a t a Kamada, Seiichi, 1964Braid and knot theory in dimension four / Seiichi Kamada. p. cm. — (Mathematical surveys and monographs ; v. 95) Includes bibliographical references and index. ISBN 0-8218-2969-6 (alk. paper) 1. Braid theory. 2. Knot theory. I. Title. II. Mathematical surveys and monographs ; no. 95. QA612.23.K36 2002 514.224—dc21 2002018274 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, P. O. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to reprint-permissionOams.org. © 2002 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at URL: http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 07 06 05 04 03 02