Linking numbers in rational homology 3-spheres, cyclic branched covers and infinite cyclic covers

Abstract

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 $\mathbb{Q}$ and in $Q(\mathbb{Z}[t,t^{-1}])$ respectively, where $Q(\mathbb{Z}[t,t^{-1}])$ denotes the quotient field of $\mathbb{Z}[t,t^{-1}]$. It is known that the modulo-$\mathbb{Z}$ linking number in the rational homology 3-sphere is determined by the linking matrix of the framed link and that the modulo-$\mathbb{Z}[t,t^{-1}]$ 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 $\mathbb{Z}$' and ` modulo $\mathbb{Z}[t,t^{-1}]$'. 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.

LaTeX, 24 pages, 6 figures