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In this chapter we recollect the preliminary materials from topology and number theory. In particular, we give a summary about fundamental groups and Galois theory for topological spaces and arithmetic rings, together with the basic concepts and examples in 3-dimensional topology and number fields. We also review class field theory as arithmetic duality theorems in Galois, etale cohomology groups.

In this chapter we discuss close parallels between Alexander-Fox theory and Iwasawa theory such as the Alexander polynomial and the Iwasawa polynomial, based on the analogy between the infinite cyclic covering of a knot exterior and the cyclotomic ℤ p -extension of a number field. In particular, we give asymptotic formulas on the orders of the 1st homology groups (p-ideal class groups) of cyclic ramified coverings (extensions).

In this chapter we review Hilbert theory which deals with, in a group-theoretic manner, the decomposition of a prime in a finite Galois extension of number fields. Based on the analogies in Chap. 3, we give a topological analogue of Hilbert theory for a finite Galois covering of 3-manifolds.

In this chapter we introduce the differential module for a group homomorphism and show the Crowell exact sequence associated to a short exact sequence of groups. Applying these constructions to the Abelianization map of a link group, we obtain the Alexander module of a link and the exact sequence relating the Alexander module with the link module. We then discuss analogous constructions for pro-finite (pro-l) groups to obtain the complete differential module and the complete Crowell exact sequence. Applying these constructions to a homomorphism from a Galois group with restricted ramification, we obtain the complete Alexander module for primes and the exact sequence relationg the complete Alexander module with a Galois (Iwasawa) module.

In this chapter we discuss the analogy between Galois groups with restricted ramification and link groups. In particular, we shall see the close analogy between Milnor’s theorem on the structure of a link group and a theorem by H. Koch on the structure of a pro-l Galois group over the rational number field with restricted ramification.

In this chapter we take up the Milnor invariants, higher order linking numbers, of a link in the 3-sphere, together with the Fox free differential calculus as a basic tool. By the analogy between a link group and a Galois group with restricted ramification in Chap. 7, we introduce arithmetic analogues of the Milnor invariants for prime numbers by using the pro-l Fox differential calculus, and show that they may be regarded as multiple generalization of the power residue symbol and the Redei triple symbol.

In this final chapter, as an extension of the analogies in Chap. 13 to 2-dimensional representations, we discuss intriguing analogies between the family of holonomy representations associated to deformations of hyperbolic structures (due to W. Thurston) and the family of Galois representations associated to deformations of p-adic ordinary modular forms (due to H. Hida and B. Mazur).

In this chapter, in view of the analogy between a knot group and a prime group, we discuss some analogies between the moduli spaces of 1-dimensional representations of knot and prime groups, and the associated higher Alexander and Iwasawa invariants.

In this chapter we first study the 2-part of the 1st homology group of a double covering of the 3-sphere ramified over a link, introducing the higher order linking matrices which are defined by using the Milnor numbers of the link in Chap. 8. Imitating the method for a link, we study the 2-part of the narrow ideal class group of a quadratic extension of the rationals, using the arithmetic Milnor numbers introduced in Chap. 8. Our theorem may be regarded as a higher order generalization of Gauss’ and Redei’s theorems on the 2-rank and 4-rank of the ideal class group.

In this chapter we explain the basic analogies between knots and primes, 3-manifolds and number rings and present a dictionary of these analogies, which will be fundamental and used throughout subsequent chapters.

In this chapter we discuss the close analogy between the linking number and the Legendre symbol, based on the analogies between knots and primes in Chap. 3. An analogy between Gauss’ linking integral and Gauss sum is also pointed out.

In this chapter we review Gauss’ genus theory from the link-theoretic point of view. We shall see that the notion of genera is defined by using the idea analogous to the linking number. We also present, vice versa, a topological analogue of genus theory.