A classifying invariant of knots, the knot quandle

Closed incompressible surfaces in alternating knot and link complements

Some aspects of classical knot theory

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

Braid and Knot Theory in Dimension Four

The usual algebraic construction used to study the symmetries of an object is the group of automorphisms of that object. In many geometric settings, however, one may interpret the symmetries in a more intimate manner by an algebraic structure on the object itself. Define a quandle to be a set equipped with two binary operations, (x, y) 7→ x . y and (x, y) 7→ x . y, which satisfies the axioms Q1. x . x = x. Q2. (x . y) . y = x = (x . y) . y. Q3. (x . y) . z = (x . z) .(y . z). Call the map S(y) sending x to x . y the symmetry at y. To each point y of a symmetric space there is a symmetry S(y) of the space. By defining x . y = x . y to be the image of x under S(y), the symmetric space becomes a quandle. Call a quandle satisfying x . y = x . y an involutory quandle. Loos [1] has defined a symmetric space as a manifold with an involutory quandle structure such that each point y is an isolated fixed point of S(y). The underlying set of a group G along with the operations of conjugation, x . y = y−1xy and x . y = yxy−1 form a quandle ConjG. Moreover, the theory of conjugation may be regarded as the theory of quandles in the sense that any equation in . and . holding in ConjG for all groups G also holds in any quandle. If the center of G is trivial, then ConjG determines G. Let G be a group and n ≥ 2. The n-core of G is the set {(x1, x2, . . . , xn) ∈ G |x1x2 . . . xn = 1} along with the operation (x1, x2, . . . , xn) .(y1, y2, . . . , yn) = (y −1 n xny1, y −1 1 x1y2, . . . , y −1 n−1xn−1yn). The n-core is an n-quandle, that is, each symmetry has order dividing n. The group G is simple if and only if its n-core is a simple quandle. Let G be a noncyclic simple group and Q a nontrivial conjugacy class in H viewed as a subquandle of ConjG. Then Q is a simple quandle.

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Jonathan Simon

Papers

An Algebraic Classification of Knots in S 3