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.

of ideal classes (2.30), one finds n i ∈ℕ for each i so that , . Set . Since A is a localization of , A is a Dedekind domain and . Choose an algebraically closed field Ω containing k which defines a base point . For a finite extension K/k in Ω, if any maximal ideal which is not contained in S is unramified in K/k, we say that K/k is unramified outside ( being the set of infinite primes of k). Let be the composite field of all finite Galois extensions k i of k in Ω which are unramified outside .

we have Here Br(R) stands for the Brauer group of R [NSW, Chap. VI, Sect. 3] and n A:={x∈A∣nx=0} for an Abelian (additive) group A. Since H 1(k,ℤ/nℤ)∗=Gal (k ab/k)/nGal (k ab/k) and the localization map is injective (Hasse principle for the Brauer group [ibid, Chap. VIII, Sect. 1]), the Tate–Poitou exact sequence yields the following isomorphism Taking the projective limit , we obtain the reciprocity homomorphism of class field theory (2.7) The map ρ k is surjective and Ker (ρ k )

Masanori Morishita Knots and PrimesAn Introduction to Arithmetic Topology Masanori MorishitaGraduate School of Mathematics, Kyushu University, Fukuoka 819-0395, Japan morisita@math.kyushu-u.ac.jp ISSN 0172-5939e-ISSN 2191-6675 ISBN 978-1-4471-2157-2e-ISBN 978-1-4471-2158-9 Springer London Dordrecht Heidelberg New York British Library Cataloguing in Publication Data A catalogue record for this book is available from the British LibraryLibrary of Congress Control Number: 2011940954 �

(Example 2.6). The knot group G K is the object which reflects how K is knotted in M. In fact, it is known that a prime knot K in S 3 is determined by the knot group G K . Namely, for prime knots K,L⊂S 3, we have the following [GL, Wh]: (3.7) Here K≃L means that there is an auto-homeomorphism h of S 3 such that h(K)=L. Let α and β be a meridian and a longitude of K respectively. The image of the homomorphism π 1(∂X K )=〈α,β | [α,β]=1〉→G K induced by the inclusion ∂X K ↪X K is called the

cyclotomic ℤ p -extension of ℚ (Example 2.46). We fix a topological generator γ of Gal (ℚ∞/ℚ)=1+pℤ p and identify ℤ p [[Gal (ℚ∞/ℚ)]] with by the correspondence γ↔1+T. We choose τ∈G {p} which is sent to γ under the natural homomorphism G {p}→Gal (ℚ∞/ℚ). The following theorem may be regarded as an arithmetic analogue for of Theorem 14.1.1 Theorem 14.8 ([MT1]) The map is p-adically bianalytic in a neighborhood of φ ∘. Here φ ∘ is defined by . Proof Since is p-ordinary for , we have