Algebraic Structure of Knot Modules
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772 Jerome P. Levine
Algebraic Structure of Knot Modules II
I
III II
I III
Author Jerome P. Levine Department of Mathematics Brandeis University Waltham, MA 02154 USA
AMS Subject Classifications (1980): 13C05, 57 Q 4 5
ISBN 3 - 5 4 0 - 0 9 ? 3 9 - 2 Springer-Verlag Berlin Heidelberg NewYork ISBN 0 - 3 8 ? - 0 9 ? 3 9 - 2 Springer-Verlag NewYork Heidelberg Berlin Library of Congress Cataloging in PublicationData Levine,JeromeP 193?Algebraic structure of knot modules. (Lecture notes in mathematics;772) Bibliography: p. Includes index. 1. Knot theory. 2. Modules(Algebra)3. Invariants. I. Title. II. Series: Lecture notes in mathematics(Berlin); 772. QA3.L28 no. 772 [QA612.2] 510s [514'.224] 80-246 ISBN 0-38?-09?39-2 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1980 Printed in Germany
TABLE
OF
CONTENTS
Introduction i.
The
derived
exact
2.
Finite
3.
Realization
4.
A. of i
5.
Product
6.
Classification
7.
Rational
8.
Z-torsion-free
9.
H-only
sequences
1 4
modules of
finite
finite
modules
6 8
modules
structure
on
of
finite
derived
modules
product
9 16
structure
18
invariants modules
20
torsion
21
I0.
Statement
of
realization
ii.
Inductive
construction
12.
Inductive
recovery
of
13.
Homogeneous
and
elementary
14.
Realization
of
15.
Classification
16.
Completion
17.
Classification
of
18.
Classification
fails
i9.
Product
20.
Classification
21.
Realization
22.
Product
of
of
derived
derived
sequences
24 26
sequences
32
modules
elementary of
23
theorem
34
modules
elementary
36
modules
proof
structure
structure
H-primary
on of
of
39
in
product on
degree
H-primary
product
40
modules 4
46
modules
48 53
structure structure
semi-homogeneous
on
homogeneous modules
modules
59 68
JV
23.
A
non-semi-homogeneous
24.
Rational
25.
Non-singular
26.
Norm
27.
Dedekind
28.
A
29.
Computation
30.
Determination
of
ideal
class
31.
The
symmetric
case
classification
for
quadratic
over
reduction
domain
lattice
75 85 88 90 92
criterion
low-degree
structure
a Dedekind
p-adic
Dedekind of
product
a non-singular
criterion:
computable
References
of
lattices
criterion
70
module
cases group
95 96 98 102
INTRODUCTION In the study of n-dimensional spheres
in (n + 2)-space,
A-modules
AI,
..., A n
A = Z[t, t-l], These modules
knots,
one encounters
(the Alexander
i.e.
imbedded n-
a collection
modules),
of
where
the ring of integral Laurent polynomials. encompass
many of the classical
knot invari-
ants. T
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