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Knots and Primes

An Introduction to Arithmetic Topology

  • Textbook
  • © 2012

Overview

Authors:
  1. Masanori Morishita

    1. Graduate School of Mathematics, Kyushu University, Fukuoka, Japan

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  • Starts at an elementary level and builds up to a more advanced theoretical discussion

  • Written by a world expert on arithmetic topology

  • A large number of illustrative examples are provided throughout?

  • Includes supplementary material: sn.pub/extras

Part of the book series: Universitext (UTX)

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Table of contents (14 chapters)

  1. Front Matter

    Pages I-XI
    Download chapter PDF

  2. Introduction

    • Masanori Morishita
    Pages 1-7

  3. Preliminaries—Fundamental Groups and Galois Groups

    • Masanori Morishita
    Pages 9-48

  4. Knots and Primes, 3-Manifolds and Number Rings

    • Masanori Morishita
    Pages 49-53

  5. Linking Numbers and Legendre Symbols

    • Masanori Morishita
    Pages 55-60

  6. Decompositions of Knots and Primes

    • Masanori Morishita
    Pages 61-68

  7. Homology Groups and Ideal Class Groups I—Genus Theory

    • Masanori Morishita
    Pages 69-76

  8. Link Groups and Galois Groups with Restricted Ramification

    • Masanori Morishita
    Pages 77-84

  9. Milnor Invariants and Multiple Residue Symbols

    • Masanori Morishita
    Pages 85-109

  10. Alexander Modules and Iwasawa Modules

    • Masanori Morishita
    Pages 111-124

  11. Homology Groups and Ideal Class Groups II—Higher Order Genus Theory

    • Masanori Morishita
    Pages 125-139

  12. Homology Groups and Ideal Class Groups III—Asymptotic Formulas

    • Masanori Morishita
    Pages 141-149

  13. Torsions and the Iwasawa Main Conjecture

    • Masanori Morishita
    Pages 151-159

  14. Moduli Spaces of Representations of Knot and Prime Groups

    • Masanori Morishita
    Pages 161-169

  15. Deformations of Hyperbolic Structures and p-Adic Ordinary Modular Forms

    • Masanori Morishita
    Pages 171-179

  16. Back Matter

    Pages 181-191
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Keywords

About this book

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. ​

Reviews

“This is one of the best textbook I have seen in the last few years. … this books is amazing! I really enjoyed it and I hope you will also enjoy it. It definitely should be part of your library if you work in number theory and/or topology. This book will become a classical very soon!” (Philosophy, Religion and Science Book Reviews, bookinspections.wordpress.com, June, 2016)

“The book under review is the first systematic treatment of the subject in a format suitable for a textbook. … The book is rich in material for anybody interested in either the arithmetic or the topological side, and the connections and interactions are presented in a very convincing and detailed way.” (Matilde Marcolli, Mathematical Reviews, March, 2013)

“Once you’ve lived long enough in mathematics, the themes addressed in Knots and Primes: An Introduction to Arithmetic Topology are both familiar and exceedingly attractive. This is a fascinating topic … and Morishita’s book is an important contribution. … it will spur a lot of work in this beatiful and accessible area of contemporary mathematics.” (Michael Berg, The Mathematical Association of America, May, 2012)

Authors and Affiliations

  • Graduate School of Mathematics, Kyushu University, Fukuoka, Japan

    Masanori Morishita

Bibliographic Information

  • Book Title: Knots and Primes

  • Book Subtitle: An Introduction to Arithmetic Topology

  • Authors: Masanori Morishita

  • Series Title: Universitext

  • DOI: https://doi.org/10.1007/978-1-4471-2158-9

  • Publisher: Springer London

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer-Verlag London Limited 2012

  • Softcover ISBN: 978-1-4471-2157-2Published: 29 November 2011

  • eBook ISBN: 978-1-4471-2158-9Published: 27 November 2011

  • Series ISSN: 0172-5939

  • Series E-ISSN: 2191-6675

  • Edition Number: 1

  • Number of Pages: XI, 191

  • Number of Illustrations: 42 b/w illustrations

  • Additional Information: English Language edition of 'Musubime to Sosu' Copyright © Springer Japan 2009

  • Topics: Number Theory, Topology, Mathematics, general

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