Linking Numbers in Rational Homology 3-spheres, Cyclic Branched Covers and Infinite Cyclic Covers

We study the linking numbers in a rational homology 3-sphere and in the infinite cyclic cover of the complement of a knot. They take values in Q and inQ(Z[t, t−1]) respectively, where Q(Z[t, t−1]) denotes the quotient field of Z[t, t−1]. It is known that the modulo-Z linking number in the rational homology 3-sphere is determined by the linking matrix of the framed link and that the modulo-Z[t, t−1] linking number in the infinite cyclic cover of the complement of a knot is determined by the Seifert matrix of the knot. We eliminate ‘modulo Z’ and ‘modulo Z[t, t−1]’. When the finite cyclic cover of the 3-sphere branched over a knot is a rational homology 3-sphere, the linking number of a pair in the preimage of a link in the 3-sphere is determined by the Goeritz/Seifert matrix of the knot. 2000 Mathematics Subject Classification. Primary 57M25; Secondary 57M10, 57M12