Knots and Primes

1. Front Matter

   Pages I-XI

   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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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. ​

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

“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)

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

  Masanori Morishita

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