# Knots and Primes

## An Introduction to Arithmetic Topology

• Masanori Morishita
Book

Part of the Universitext book series (UTX)

1. Front Matter
Pages I-XI
2. Masanori Morishita
Pages 1-7
3. Masanori Morishita
Pages 9-48
4. Masanori Morishita
Pages 49-53
5. Masanori Morishita
Pages 55-60
6. Masanori Morishita
Pages 61-68
7. Masanori Morishita
Pages 69-76
8. Masanori Morishita
Pages 77-84
9. Masanori Morishita
Pages 85-109
10. Masanori Morishita
Pages 111-124
11. Masanori Morishita
Pages 125-139
12. Masanori Morishita
Pages 141-149
13. Masanori Morishita
Pages 151-159
14. Masanori Morishita
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15. Masanori Morishita
Pages 171-179
16. Back Matter
Pages 181-191

### Introduction

This is a foundation for arithmetic topology - a new branch of mathematics which is focused upon the analogy between knot theory and number theory.

Starting with an informative introduction to its origins, namely Gauss, this text provides a background on knots, three manifolds and number fields. Common aspects of both knot theory and number theory, for instance knots in three manifolds versus primes in a number field, are compared throughout the book. These comparisons begin at an elementary level, slowly building up to advanced theories in later chapters. Definitions are carefully formulated and proofs are largely self-contained.

When necessary, background information is provided and theory is accompanied  with a number of useful examples and illustrations, making this a useful text for both undergraduates and graduates in the field of knot theory, number theory and geometry.

### Keywords

3-manifolds arithmetic topology homology groups knots and primes legendre symbols number rings

#### Authors and affiliations

• Masanori Morishita
• 1
1. 1.Graduate School of MathematicsKyushu UniversityFukuokaJapan

### Bibliographic information

• DOI https://doi.org/10.1007/978-1-4471-2158-9
• Copyright Information Springer-Verlag London Limited 2012
• Publisher Name Springer, London
• eBook Packages Mathematics and Statistics
• Print ISBN 978-1-4471-2157-2
• Online ISBN 978-1-4471-2158-9
• Series Print ISSN 0172-5939
• Series Online ISSN 2191-6675