Morishita - Knots and primes: an introduction to arithmetic topology

Masanori Morishita 1

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Abstract

Starting with the work of Gauss on quadratic residues and linking numbers, we review some histories of knot theory and number theory that branched out after Gauss. In particular, we trace the string of thoughts on geometrization of number theory which led to the theme of this book, arithmetic topologya new branch of mathematics bridging between knot theory and number theory. An outline of this book is also included.

In his youth, C.F. Gauss proved the law of quadratic reciprocity and further created the theory of genera for binary quadratic forms ([], 1801).

For an odd prime number p and an integer a prime to p , consider the quadratic equation modulo p : According as this equation has an integral solution or not, the integer a is called a quadratic residue or quadratic non-residue mod p , and the Legendre symbol is defined by For odd prime numbers p and q , Gauss proved the following relation between p being a quadratic residue mod q and q being a quadratic residue mod p : In particular, the symmetric relation holds if p or q 1mod4:

In terms of algebra today, Gauss genus theory may be viewed as a classification theory of ideals of a quadratic field . Let be the ring of integers of k . Nonzero fractional ideals and of (i.e., finitely generated -submodules of k ) are said to be in the same class in the narrow sense if there is k such that where denotes the conjugate of . Let H +( k ) denote the set of these classes. For the sake of simplicity, we assume for the moment that m = p 1 p r ( p 1,, p r being different prime factors) and p i 1mod4 (1 i r ). Note that in each class we may choose an ideal in prime to m . Such ideals and are said to be in the same genus, written by , if one has where . This gives a well-defined equivalence relation on H +( k ) and we can classify H +( k ) by the relation . Gauss proved that H +( k ) forms a finite Abelian group by the multiplication of fractional ideals, which is called the ideal class group in the narrow sense, that the correspondence gives rise to the following isomorphism and hence that the number of genera is 2 r 1. Gauss investigation on quadratic residues may be seen as an origin of the modern development of algebraic number theory.

On the other hand, in [] 1833, Gauss discovered the notion of the linking number, together with its integral expression, in the course of his investigations of electrodynamics. Let K and L be disjoint, oriented simple closed curves in 3 (i.e., a 2-component link) with parametrizations given by smooth functions a :[0,1]3 and b :[0,1]3, respectively. Let us turn on an electric current with strength I in L so that the magnetic field B ( x ) ( x 3) is generated. By the law of BiotSavart, B ( x ) is given by where 0 stands for the magnetic permeability of a vacuum. Then Gauss showed the following integral formula: namely, Here lk( L , K ) is an integer, called the linking number of K and L , which is defined as follows. Let L be an oriented surface with L = L . We may assume that K crosses L at right angles. Let P be an intersection point of K and L . According as a tangent vector of K at P has the same or opposite direction to a normal vector of L at P , we assign a number ( P ):=1 or 1 to each P : Let P 1,, P m be the set of intersection points of K and L . Then the linking number lk( L , K ) is defined by By this definition or by Gauss integral formula, we easily see that the symmetric relation holds: