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.
Preliminaries - Fundamental Groups and Galois Groups.- Knots and Primes, 3-Manifolds and Number Rings.- Linking Numbers and Legendre Symbols.- Decompositions of Knots and Primes.- Homology Groups and Ideal Class Groups I - Genus Theory.- Link Groups and Galois Groups with Restricted Ramification.- Milnor Invariants and Multiple Power Residue Symbols.- Alexander Modules and Iwasawa Modules.- Homology Groups and Ideal Class Groups II - Higher Order Genus Theory.- Homology Groups and Ideal Class Groups III - Asymptotic Formulas.- Torsions and the Iwasawa Main Conjecture.- Moduli Spaces of Representations of Knot and Prime Groups.- Deformations of Hyperbolic Structures and of p-adic Ordinary Modular Forms.