Link Theory in Manifolds

Any topological theory of knots and links should be based on simple ideas of intersection and linking. In this book, a general theory of link bordism in manifolds and universal constructions of linking numbers in oriented 3-manifolds are developed. In thi

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Uwe Kaiser

Link Theory in Manifolds

Springer

Author Uwe Kaiser Fachbereich Mathematik Universitat-Gesamthochschule Siegen Holderlinstralle 3 D-57068 Siegen, Germany E-mail: [email protected]

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Die Deutsche Bibliothek - CIP-Einheitsaufnahme

Kaiser, Uwe:

Link theory in manifolds / Uwe Kaiser. - Berlin; Heidelberg; New York; Barcelona; Budapest; Hong Kong; London; Milan; Paris; Santa Clara ; Singapore ; Tokyo : Springer, 1997 (Lecture notes in mathematics; 1669) ISBN 3-540-63435-5

Mathematics Subject Classification (I 991): 57M25, 57N 10 ISSN 0075-8434 ISBN 3-540-63435-5 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1997 Printed in Germany The use of general descriptive names, registered names, trademarks, 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. Typesetting: Camera-ready TEX output by the author SPIN: 10553330 46/3142-543210 - Printed on acid-free paper

TO MY FATHER

INTRODUCTION

Invariants of links in 3-manifolds have been defined and studied classically through algebraic topology. In the last decade ideas from singularity theory, quantum field theory and statistical mechanics gave rise to the new theories of Vassiliev- and quantum invariants. But the resulting combinatorial context and the topology of links are not nicely combined. The Conway polynomial V'K(Z) of links K in 53 is an invariant, which is well understood from both the classical and the modern viewpoint. It is combinatorially characterized by V'u nknot = 1 and the Conway relation

Here

x x «.

)(

are three links, which differ only in a 3-ball in the indicated way. While this characterization is simple it is extremely difficult to be used for an existence proof [Kal]. This is quite in contrast to the Jones polynomial [Ka2], which is the prominent example of the recent invariants. In the early seventies John Conway suggested to generalize the Conway polynomial for the case of oriented 3-manifolds M in the following way: Consider the quotient of the free Z[z]-module on the set of isotopy classes of oriented links in M by the submodule generated by all elements K; -f{_ -zf{o. Obviously the re